# General representation theory in relatively closed monoidal categories

Alexey Silantyev<EMAIL_ADDRESS>
###### Abstract

We apply the notion of relative adjoint functor to generalise closed monoidal
categories. We define representations in such categories and give their
relation with left actions of monoids. The translation of these
representations under lax monoidal functors is investigated. We introduce
tensor product of representations of bimonoids as a functorial binary
operation and show how symmetric lax monoidal functors act on this product.
Finally we apply the general theory to classical and quantum representations.

Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia

State University “Dubna”, 141980 Dubna, Moscow region, Russia

Keywords: general representation theory, relative adjoints; monoidal
categories, quadratic algebras, quantum representations.

###### Contents

  1. 1 Introduction
  2. 2 Relative adjunctions and relatively closed monoidal categories
    1. 2.1 Notations and conventions
    2. 2.2 Corollaries of Yoneda Lemma
    3. 2.3 Relative right and left adjoint functors
    4. 2.4 Relatively closed monoidal categories
  3. 3 General representation theory
    1. 3.1 Representations in relatively closed monoidal categories
    2. 3.2 Tensor product of representations
    3. 3.3 Translation of representations by a functor
  4. 4 Examples
    1. 4.1 Representations on vector spaces
    2. 4.2 Quantum representations on quadratic algebras
  5. Conclusion

##  1 Introduction

The classical representation theory deals with representations of algebras,
groups and other algebraic objects which can be represented by linear
operators on vector spaces. An idea of quantum representations goes back to
the works of Yuri Manin [Man87, Man88, Man91], where he studied the category
of finitely generated quadratic algebras. He proposed to regard the objects of
the opposite category as a ‘non-commutative’ (or ‘quantum’) generalisation of
finite-dimensional vector spaces and called them quantum linear spaces.

In [S21] we introduced a notion of quantum representation, which generalises
finite-dimensional representations of finite-dimensional algebras and
algebraic groups to the case of quantum linear representation space. To define
and investigate quantum representations we used there a general approach to
representations in monoidal categories. In the present paper we continue to
develop this approach, which we call general representation theory.

A representation of an algebra ${\mathfrak{A}}$ on a vector space $V$ can be
regarded in two ways: as an action $a\colon{\mathfrak{A}}\otimes V\to V$ or as
an algebra homomorphism $\rho\colon{\mathfrak{A}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$, where $\mathop{\mathrm{\bf end}}\nolimits(V)$ is the
algebra of linear operators on $V$. We use the word ‘representation’ for such
morphisms $\rho$. This is a basic example of the general notion of
representation in a monoidal category for the category of vector spaces, where
the role of the monoidal product is played by the tensor product $\otimes$.

The notion of action is generalised for any (symmetric) monoidal category
$({\mathbf{C}},\otimes)$ as an action of a monoid (we consider the case of
symmetric monoidal categories only). This notion is equivalent to the notion
of representation iff $({\mathbf{C}},\otimes)$ is closed. In this case we can
use the internal hom-functor $\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{C}}$
to define representations and relate them with the (left) actions. In [S21, §
3] we considered more general situation, when the functor $-\otimes
V\colon{\mathbf{C}}\to{\mathbf{C}}$ has a right adjoint for each
$V\in{\mathbf{P}}$, where ${\mathbf{P}}$ is a full subcategory of
${\mathbf{C}}$. We used the theorem on adjunction with a parameter [MacLane, §
4.7, Th. 3] to obtain a generalised internal hom-functor $\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{C}}$
with a natural isomorphism
$\alpha_{X,V,Z}\colon\mathop{\mathrm{Hom}}\nolimits(X\otimes
V,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
hom}}\nolimits(V,Z)\big{)}$, where $X\in{\mathbf{C}}$, $V\in{\mathbf{P}}$,
$Z\in{\mathbf{C}}$.

But it turned out that this condition is too strong. We need the isomorphisms
$\alpha_{X,V,Z}$ for $Z\in{\mathbf{P}}$ only. This can not be formulated in
terms of usual adjoint functors, so we use the notion of relative adjoints
introduced by Friedrich Ulmer in [Ulm]. We require that for each
$V\in{\mathbf{P}}$ the functor $-\otimes V\colon{\mathbf{C}}\to{\mathbf{C}}$
has a right adjoint relative to ${\mathbf{P}}$ and then we say that the
symmetric monoidal category $({\mathbf{C}},\otimes)$ is closed relative to the
parametrising subcategory ${\mathbf{P}}\subset{\mathbf{C}}$. This approach
allows us to generalise the results of [S21] to the monoidal category
${\mathbf{C}}$ opposite to the category of all the ${\mathbb{Z}}$-graded
algebras with the Manin product ‘$\circ$’ and the parametrising subcategory
${\mathbf{P}}$ consisting of the quantum linear spaces.

In the situations such as Quantum Representation Theory it is more convenient
to work in the opposite category to the category ${\mathbf{C}}$. By this
reason we double all the important results for this case by using categorical
duality: we consider monoidal categories coclosed relative to a full
subcategory, comonoids, corepresentations etc.

The paper is organised as follows. In the section 2 we generalise the notion
of (co)closed monoidal category by means of relative adjoints. Subsection 2.1
is preliminary. In Subsection 2.2 we formulate some corollaries of Yoneda
Lemma. Subsection 2.3 is devoted to the theory of relative adjunctions. In
Subsection 2.4 we define relatively (co)closed monoidal category and describe
internal (co)hom and (co)end for this case. Section 3 is devoted to
(co)representations in relatively (co)closed monoidal categories. In
Subsection 3.1 we introduce (co)representations of (co)monoids, their
morphisms and give their relation with (co)actions. In Subsection 3.2 we
describe tensor product of (co)representations of bimonoids. The translation
of the (co)representations and their tensor products under monoidal functors
is investigated in Subsection 3.3. Then, in Section 4, we give the basic
examples. Classical representations of various types of groups, algebras and
their mixed (semi-linear) versions is described in Subsection 4.1 in terms of
the general representation theory. In Subsection 4.2 we apply the general
theory to the quantum representations.

Acknowledgements. The author thanks V. Rubtsov for useful advice.

##  2 Relative adjunctions and relatively closed monoidal categories

Before construction of the general representation theory we need to introduce
some types of monoidal categories which is appropriate for definition of
(co)representations. The condition on such monoidal categories is given by
means of the notion of relative adjoints (with a parameter). For completeness
we formulate and prove some needed statements on the relative adjoints. This
requires in turn some facts following from Yoneda Lemma.

###  2.1 Notations and conventions

Fist of all, let us fix some conventions. In this work we use the terms and
notations introduced in [S21, § 2]. Let us remind some of them. We also recall
some definitions from [MacLane] and introduce some terminology used below.

2.1.1. Morphisms and functors. In order to unify the notations with the
previous publications we denote the composition of morphisms $f,g$ in a
category ${\mathbf{C}}$ by $f\cdot g$ instead of $f\circ g$. For a fixed
object $X\in{\mathbf{C}}$ denote by
$\mathop{\mathrm{Hom}}\nolimits(X,-)\colon{\mathbf{C}}\to{\mathbf{Set}}$ the
functor from ${\mathbf{C}}$ to the category of sets, it maps an object
$Y\in{\mathbf{C}}$ to the set
$\mathop{\mathrm{Hom}}\nolimits(X,Y)=\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}(X,Y)\in{\mathbf{Set}}$
which consists of all the morphisms in ${\mathbf{C}}$ of the form $X\to Y$, a
morphism $f\colon Y\to Z$ is translated by this functor to the map
$\mathop{\mathrm{Hom}}\nolimits(X,f)\colon\mathop{\mathrm{Hom}}\nolimits(X,Y)\to\mathop{\mathrm{Hom}}\nolimits(X,Z)$,
$g\mapsto f\cdot g\in\mathop{\mathrm{Hom}}\nolimits(X,Z)$,
$\;\forall\,g\in\mathop{\mathrm{Hom}}\nolimits(X,Y)$. Analogously we define
the contravariant functor
$\mathop{\mathrm{Hom}}\nolimits(-,X)\colon{\mathbf{C}}\to{\mathbf{Set}}$,
$\mathop{\mathrm{Hom}}\nolimits(f,X)\colon\mathop{\mathrm{Hom}}\nolimits(Z,X)\to\mathop{\mathrm{Hom}}\nolimits(Y,X)$,
$h\mapsto h\cdot f\in\mathop{\mathrm{Hom}}\nolimits(Y,X)$,
$h\in\mathop{\mathrm{Hom}}\nolimits(Z,X)$. Together these formulae give the
bifunctor
$\mathop{\mathrm{Hom}}\nolimits=\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}=\mathop{\mathrm{Hom}}\nolimits(-,-)\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{Set}}$.

Let $F\colon{\mathbf{C}}\to{\mathbf{D}}$ be a functor from a category
${\mathbf{C}}$ to a category ${\mathbf{D}}$. If
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ is a subcategory, then we can
restrict $F$ to ${\mathbf{C}}^{\prime}$ and get a new functor
${\mathbf{C}}^{\prime}\to{\mathbf{D}}$, which we usually denote by the same
letter $F$; formally, this is a composition $F\cdot
E_{{\mathbf{C}}^{\prime}}$, where
$E_{{\mathbf{C}}^{\prime}}\colon{\mathbf{C}}^{\prime}\hookrightarrow{\mathbf{C}}$
is the categorical embedding. If ${\mathbf{D}}^{\prime}$ is a subcategory of
${\mathbf{D}}$ such that $Ff$ is a morphism in ${\mathbf{D}}^{\prime}$ for any
morphism $f$ in ${\mathbf{C}}^{\prime}$, then the restriction gives a functor
$F\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}^{\prime}$.

2.1.2. Natural transformations. Let $\alpha\colon F\to G$ be a natural
transformation between functors $F\colon{\mathbf{C}}\to{\mathbf{D}}$ and
$G\colon{\mathbf{C}}\to{\mathbf{D}}$. We sometimes omit the subscript $X$ of
the components $\alpha_{X}\colon FX\to GX$. For instance, for a morphism
$f\colon Y\to Z$ in ${\mathbf{C}}$ the formulae $f_{*}(g)=f\cdot g$ and
$f^{*}(h)=h\cdot f$ define the components
$f_{*}=(f_{*})_{X}=\mathop{\mathrm{Hom}}\nolimits(X,f)\colon\mathop{\mathrm{Hom}}\nolimits(X,Y)\to\mathop{\mathrm{Hom}}\nolimits(X,Z)$
and
$f^{*}=(f^{*})_{X}=\mathop{\mathrm{Hom}}\nolimits(f,X)\colon\mathop{\mathrm{Hom}}\nolimits(Z,X)\to\mathop{\mathrm{Hom}}\nolimits(Y,X)$
of the natural transformations
$f_{*}\colon\mathop{\mathrm{Hom}}\nolimits(-,Y)\to\mathop{\mathrm{Hom}}\nolimits(-,Z)$
and
$f^{*}\colon\mathop{\mathrm{Hom}}\nolimits(Z,-)\to\mathop{\mathrm{Hom}}\nolimits(Y,-)$
respectively.

For natural transformations $\alpha\colon F\to G$ and $\beta\colon G\to H$
between functors $F,G,H\colon{\mathbf{C}}\to{\mathbf{D}}$ we denote their
composition by $\beta\cdot\alpha\colon F\to H$. This is a natural
transformation with the components
$(\beta\cdot\alpha)_{X}=\beta_{X}\cdot\alpha_{X}\colon FX\to HX$.

We say that the morphisms $\alpha=\alpha_{X}\colon FX\to GX$ are natural in
$X$ iff they define a natural transformation $\alpha\colon F\to G$. Recall
that the natural transformation $\alpha\colon F\to G$ is an isomorphism in the
category of functors ${\mathbf{C}}\to{\mathbf{D}}$ iff all the components
$\alpha=\alpha_{X}\colon FX\to GX$ are isomorphisms in ${\mathbf{D}}$; the
inverse natural transformation $\alpha^{-1}\colon G\to F$ has components
$\alpha^{-1}=(\alpha_{X})^{-1}\colon GX\to FX$ inverse to $\alpha_{X}$. Let
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ be a subcategory, then we say that
$\alpha=\alpha_{X}\colon FX\to GX$ are natural in $X\in{\mathbf{C}}^{\prime}$
iff they define a natural transformation between the restricted functors $F$
and $G$, i.e. $\alpha\colon F\cdot E_{{\mathbf{C}}^{\prime}}\to G\cdot
E_{{\mathbf{C}}^{\prime}}$. In this case it is enough to suppose that the
morphisms $\alpha_{X}$ are defined for $X\in{\mathbf{C}}^{\prime}$.

Let ${\mathbf{P}}$ be also a category and $F$ and $G$ be bifunctors
${\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$. Let
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ and
${\mathbf{P}}^{\prime}\subset{\mathbf{P}}$ be subcategories. Then the
requirement that $\alpha=\alpha_{X,Y}\colon F(X,Y)\to G(X,Y)$ are natural in
$X\in{\mathbf{C}}^{\prime}$ and $Y\in{\mathbf{P}}^{\prime}$ means that these
morphisms give a natural transformation between the restrictions of $F$ and
$G$ to the subcategory
${\mathbf{C}}^{\prime}\times{\mathbf{P}}^{\prime}\subset{\mathbf{C}}\times{\mathbf{P}}$.
For a fixed object $Y\in{\mathbf{P}}$ we have functors
$F_{Y},G_{Y}\colon{\mathbf{C}}\to{\mathbf{D}}$ such that $F_{Y}(f)=F(f,Y)$ and
$G_{Y}(f)=G(f,Y)$ for any morphism $f$ in ${\mathbf{C}}$. These functors can
be interpreted as functors with a parameter. We say that
$\alpha=\alpha_{X,Y}\colon F_{Y}(X)\to G_{Y}(X)$ are natural in
$X\in{\mathbf{C}}^{\prime}$ iff for any $Y\in{\mathbf{P}}$ they give a natural
transformation between the restricted functors
$F_{Y},G_{Y}\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ (in this case we do
not suppose that $F_{Y}$ and $G_{Y}$ are functorial in $Y$, i.e. morphisms
$F(f,X)$ and $G(f,X)$ may be undefined). Analogous conventions are supposed
for the functors with three or more arguments.

Note that any functor $F\colon{\mathbf{C}}\to{\mathbf{D}}$ can be considered
as the natural transformation between the functors
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}$ and
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}F(-),F(-)\big{)}\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{Set}}$
with the components
$F\colon\mathop{\mathrm{Hom}}\nolimits(X,Y)\to\mathop{\mathrm{Hom}}\nolimits(FX,FY)$.
By fixing one of the arguments we obtain the natural transformation
$\mathop{\mathrm{Hom}}\nolimits(X,-)\to\mathop{\mathrm{Hom}}\nolimits\big{(}FX,F(-)\big{)}$
or
$\mathop{\mathrm{Hom}}\nolimits(-,Y)\to\mathop{\mathrm{Hom}}\nolimits\big{(}F(-),FY\big{)}$.

###  2.2 Corollaries of Yoneda Lemma

Yoneda Lemma and its corollaries are the main tool in the theory of adjoints
and their relative version.

2.2.1. Yoneda Lemma. Let $R$ be an object of a category ${\mathbf{D}}$ and
$K\colon{\mathbf{D}}\to{\mathbf{Set}}$ be a functor. Yoneda Lemma establishes
a one-to-one correspondence between the natural transformations from the
functor
$\mathop{\mathrm{Hom}}\nolimits(R,-)\colon{\mathbf{D}}\to{\mathbf{Set}}$ to
the functor $K$ and the elements of the set $KR$ (see [MacLane, § 3.2]). It
can be formulated as follows.

######  Lemma 2.1.

Any natural transformation $\alpha\colon\mathop{\mathrm{Hom}}\nolimits(R,-)\to
K$ can be written in the form $\alpha_{Z}(f)=K(f)(u)\in KZ$,
$f\in\mathop{\mathrm{Hom}}\nolimits(R,Z)$, for a unique element $u\in KR$.

We are interested in some particular cases of this Lemma and their
corollaries.

2.2.2. The case $K=\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$. Let
$G\colon{\mathbf{D}}\to{\mathbf{C}}$ be a functor and $X\in{\mathbf{C}}$.
Consider the functor $K=\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$,
it maps an object $Z\in{\mathbf{D}}$ to the set
$\mathop{\mathrm{Hom}}\nolimits(X,GZ)$ and a morphism $f\colon Z\to
Z^{\prime}$ to the map
$(Gf)_{*}\colon\mathop{\mathrm{Hom}}\nolimits(X,GZ)\to\mathop{\mathrm{Hom}}\nolimits(X,GZ^{\prime})$.
In this case Yoneda Lemma 2.1 takes the following form.

######  Corollary 2.2.

Any maps
$\alpha_{Z}\colon\mathop{\mathrm{Hom}}\nolimits(R,Z)\to\mathop{\mathrm{Hom}}\nolimits(X,GZ)$
natural in $Z\in{\mathbf{D}}$ have the form $\alpha(f)=(Gf)\cdot\gamma$ for a
unique morphism $\gamma\colon X\to GR$.

Denote the natural transformation
$\alpha\colon\mathop{\mathrm{Hom}}\nolimits(R,-)\to\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$
given by a morphism $\gamma\colon X\to GR$ as in Corollary 2.2 by
$\alpha=\gamma^{*}\cdot G$, this is a composition of natural transformations
in the sense of p. 2.1, where the functor $G$ is regarded as the natural
transformation
$\mathop{\mathrm{Hom}}\nolimits(R,-)\to\mathop{\mathrm{Hom}}\nolimits\big{(}GR,G(-)\big{)}$
(this is not ‘horizontal’ composition defined in [MacLane, § 2.5]). If
${\mathbf{C}}={\mathbf{D}}$ and
$G=\mathop{\mathrm{id}}\nolimits_{{\mathbf{C}}}$, then $\alpha=\gamma^{*}$.

Consider a category ${\mathbf{C}}^{\prime}$ and functors
$F_{1}\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$,
$F_{2}\colon{\mathbf{C}}^{\prime}\to{\mathbf{C}}$.

######  Corollary 2.3.

The morphisms $\gamma_{X}\colon F_{2}X\to GF_{1}X$ are natural in
$X\in{\mathbf{C}}^{\prime}$ iff the morphisms
$\alpha_{X,Z}=\gamma_{X}^{*}\cdot
G\colon\mathop{\mathrm{Hom}}\nolimits(F_{1}X,Z)\to\mathop{\mathrm{Hom}}\nolimits(F_{2}X,GZ)$
are natural in $X\in{\mathbf{C}}^{\prime}$.

Proof. If $\gamma_{X}\colon F_{2}X\to GF_{1}X$ are natural in $X$, then for
any morphism $g\colon X\to Y$ in ${\mathbf{C}}^{\prime}$ we have
$\gamma_{Y}\cdot F_{2}g=GF_{1}g\cdot\gamma_{X}$, hence
$(F_{2}g)^{*}\cdot\gamma_{Y}^{*}\cdot G=(\gamma_{Y}\cdot F_{2}g)^{*}\cdot
G=(GF_{1}g\cdot\gamma_{X})^{*}\cdot G=\gamma_{X}^{*}\cdot G\cdot(F_{1}g)^{*}$.
This implies that the diagram

(2.5)

is commutative, which means exactly that $\alpha_{X,Z}$ is natural in $X$.
Conversely, let the diagram (2.5) be commutative. Denote
$\delta_{Z}=\gamma_{X}^{*}\cdot
G\cdot(F_{1}g)^{*}=\gamma_{X}^{*}\cdot(GF_{1}g)^{*}\cdot G$, this is the
diagonal arrow
$\mathop{\mathrm{Hom}}\nolimits(F_{1}X,Z)\to\mathop{\mathrm{Hom}}\nolimits(F_{2}Y,GZ)$
in the diagram (2.5). Note that $\delta_{Z}$ are natural in $Z$, so by virtue
of Corollary 2.2 there exist unique morphisms $f_{X,Y}\colon F_{2}Y\to F_{1}X$
such that $\delta_{Z}=f_{X,Y}^{*}\cdot G$. Hence $\gamma_{Y}\cdot
F_{2}g=f_{X,Y}=GF_{1}g\cdot\gamma_{X}$. ∎

2.2.3. Universal morphisms. Recall that universal morphism from an object
$X\in{\mathbf{C}}$ to a functor $G\colon{\mathbf{D}}\to{\mathbf{C}}$ is a pair
$(R,\eta)$, where $R\in{\mathbf{D}}$ and $\eta\colon X\to GR$ are an object
and a morphism such that for any $Z\in{\mathbf{D}}$ and $f\colon X\to GZ$
there is a unique $h\colon R\to Z$ such that $f=Gh\cdot\eta$.

Analogously, universal morphism from a functor
$F\colon{\mathbf{C}}\to{\mathbf{D}}$ to an object $Z\in{\mathbf{D}}$ is a pair
$(P,\varepsilon)$, where $P\in{\mathbf{C}}$ and $\varepsilon\colon FP\to Z$
are such that for any $X\in{\mathbf{C}}$ and $h\colon FX\to Z$ there is a
unique $f\colon X\to P$ such that $h=\varepsilon\cdot Ff$. This notion can be
obtained from the previous one by the categorical duality.

In terms of p. 2.1 the condition that $(R,\eta)$ is a universal morphism from
an object $X\in{\mathbf{C}}$ to a functor $G\colon{\mathbf{D}}\to{\mathbf{C}}$
is equivalent to the bijectivity of the natural transformation $\eta^{*}\cdot
G\colon\mathop{\mathrm{Hom}}\nolimits(R,-)\to\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$.
Due to Corollary 2.2 we can reformulate it as follows (see [MacLane, § 3.2,
Prop. 1]).

######  Corollary 2.4.

Any bijections
$\mathop{\mathrm{Hom}}\nolimits(R,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}X,GZ\big{)}$
natural in $Z\in{\mathbf{D}}$ have the form $\eta^{*}\cdot G$ for a unique
universal morphism $(R,\eta)$ from $X\in{\mathbf{C}}$ to
$G\colon{\mathbf{D}}\to{\mathbf{C}}$.

######  Corollary 2.5.

Consider functors $G\colon{\mathbf{D}}\to{\mathbf{C}}$,
$F_{1}\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ and
$F_{2}\colon{\mathbf{C}}^{\prime}\to{\mathbf{C}}$. Any bijections
$\beta_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(F_{1}X,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}F_{2}X,GZ\big{)}$
natural in $X\in{\mathbf{C}}^{\prime}$ and $Z\in{\mathbf{D}}$ have the form
$\eta_{X}^{*}\cdot G$ for a unique natural transformation $\eta\colon F_{2}\to
GF_{1}$ such that $(F_{1}X,\eta_{X})$ is a universal morphism from
$F_{2}X\in{\mathbf{C}}^{\prime}$ to $G\colon{\mathbf{D}}\to{\mathbf{C}}$.

Proof. This statement follows directly from Corollaries 2.4 and 2.3. ∎

2.2.4. The case $K=\mathop{\mathrm{Hom}}\nolimits(X,-)$. If the functor $K$
has the form $K=\mathop{\mathrm{Hom}}\nolimits(X,-)$ for some
$X\in{\mathbf{D}}$, then Yoneda Lemma 2.1 is a particular case of Corollary
2.2 for ${\mathbf{C}}={\mathbf{D}}$ and
$G=\mathop{\mathrm{id}}\nolimits_{{\mathbf{D}}}$. It can be written in the
following form (see [MacLane, § 3.2]).

######  Corollary 2.6.

Any natural transformation
$\mathop{\mathrm{Hom}}\nolimits(R,-)\to\mathop{\mathrm{Hom}}\nolimits(X,-)$
has the form $\gamma^{*}$ for a unique morphism $\gamma\colon X\to R$.

Note that $(R,\gamma)$ is a universal morphism form $R$ to
$\mathop{\mathrm{id}}\nolimits_{\mathbf{D}}$ iff $\gamma$ is an isomorphism.
Hence Corollary 2.4 takes the following from.

######  Corollary 2.7.

The natural transformation
$\gamma^{*}\colon\mathop{\mathrm{Hom}}\nolimits(R,-)\to\mathop{\mathrm{Hom}}\nolimits(X,-)$
is an isomorphism of functors iff $\gamma\colon X\to R$ is an isomorphism in
${\mathbf{D}}$.

######  Remark 2.8.

The universal morphism is unique up to an isomorphism: if $(R,\eta)$ and
$(R^{\prime},\eta^{\prime})$ are two universal morphisms from $X$ to $G$, then
there is a unique isomorphism $\gamma\colon
R\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}R^{\prime}$ such that
$G\gamma\cdot\eta=\eta^{\prime}$. This fact is deduced in [MacLane, § 3.1] by
using comma categories, however it also follows from (the corollaries of)
Yoneda Lemma. To show it one needs to apply Corollaries 2.6 and 2.7 to the
composition of $(\eta^{\prime})^{*}\cdot
G\colon\mathop{\mathrm{Hom}}\nolimits(R^{\prime},-)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$
with the inverse of $\eta^{*}\cdot
G\colon\mathop{\mathrm{Hom}}\nolimits(R,-)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(-)\big{)}$
and then to take into account Corollary 2.2. Dually, a universal morphism
$(P,\varepsilon)$ from $F\colon{\mathbf{C}}\to{\mathbf{D}}$ to an object
$Z\in{\mathbf{D}}$ is unique up to a unique isomorphism.

Let $F_{1}$ and $F_{2}$ be functors ${\mathbf{C}}^{\prime}\to{\mathbf{D}}$.
Corollaries 2.2, 2.3 and 2.5 imply the following statements.

######  Corollary 2.9.

If maps
$\beta_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(F_{1}X,Z)\to\mathop{\mathrm{Hom}}\nolimits(F_{2}X,Z)$
are natural in $X\in{\mathbf{C}}^{\prime}$ and $Z\in{\mathbf{D}}$, then
$\beta_{X,Z}=\gamma_{X}^{*}$ for a unique natural transformation $\gamma\colon
F_{2}\to F_{1}$.

######  Corollary 2.10.

Any bijections
$\beta_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(F_{1}X,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(F_{2}X,Z)$
natural in $X\in{\mathbf{C}}^{\prime}$ and $Z\in{\mathbf{D}}$ have the form
$\beta_{X,Z}=\gamma_{X}^{*}$ for a unique natural isomorphism $\gamma\colon
F_{2}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F_{1}$.

###  2.3 Relative right and left adjoint functors

Now we introduce a notion of relative right/left adjunctions and its version
‘with a parameter’ which are necessary for an appropriate generalisation of
the (co)closed monoidal categories.

2.3.1. Relative adjunctions. Let ${\mathbf{C}}$ and ${\mathbf{D}}$ be
categories. Recall that a functor $F\colon{\mathbf{C}}\to{\mathbf{D}}$ is
called left adjoint for $G\colon{\mathbf{D}}\to{\mathbf{C}}$ and $G$ is called
right adjoint for $F$ when there exist bijections

$\displaystyle\alpha_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}(FX,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}(X,GZ)$
(2.6)

natural in $X\in{\mathbf{C}}$ and $Z\in{\mathbf{D}}$. It gives a natural
isomorphism $\alpha$ between the bifunctors defined as compositions

$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}F(-),-\big{)}\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{F^{\mathrm{op}}\times\mathop{\mathrm{id}}\nolimits_{\mathbf{D}}}{\mathbf{D}}^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}}{\mathbf{Set}},$
(2.7)
$\displaystyle\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}\big{(}-,G(-)\big{)}\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{\mathop{\mathrm{id}}\nolimits_{{\mathbf{C}}^{\mathrm{op}}}\times
G}{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}}{\mathbf{Set}},$
(2.8)

where
$F^{\mathrm{op}}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}^{\mathrm{op}}$
is the functor opposite to $F$. The isomorphism $\alpha$ is called adjunction
bijection for the adjoints $F,G$. The triple $(F,G,\alpha)$ is called
adjunction [MacLane, § 4.1].

Consider a full subcategory ${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$. If we
restrict the functors (2.7), (2.8) to the subcategory
${\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}^{\prime}\subset{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}$,
then we obtain isomorphisms (2.6) natural in $X\in{\mathbf{C}}$ and
$Z\in{\mathbf{D}}^{\prime}$. In this case it is sufficient to define the
functor $G$ on the subcategory ${\mathbf{D}}^{\prime}$, however the functors
$F\colon{\mathbf{C}}\to{\mathbf{D}}$ and
$G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ are not adjoint to each other,
because the domain of such $G$ does not coincide with the codomain of $F$ (we
can not reduce this situation to the adjunction case unless
$F({\mathbf{C}})\subset{\mathbf{D}}^{\prime}$, when one can consider $F$ as a
functor ${\mathbf{C}}\to{\mathbf{D}}^{\prime}$).

A generalisation of the notion of adjoint functor appropriate for this
situation was introduced by Ulmer in [Ulm, Def. 2.3]. Namely, let
${\mathbf{B}},{\mathbf{C}},{\mathbf{D}}$ be categories and
$J\colon{\mathbf{B}}\to{\mathbf{D}}$, $F\colon{\mathbf{C}}\to{\mathbf{D}}$,
$G\colon{\mathbf{B}}\to{\mathbf{C}}$ be functors. A functor $G$ is called
right adjoint for $F$ relative to $J$ iff
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}F(-),J(-)\big{)}$ and
$\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}-,G(-)\big{)}$ are
isomorphic as functors
${\mathbf{C}}^{\mathrm{op}}\times{\mathbf{B}}\to{\mathbf{Set}}$. Similarly
relative left adjoint is defined [Ulm, Def. 2.2]. In this work we consider the
case of fully faithful functor $J$ only. For this case we can interpret
${\mathbf{B}}$ as a subcategory of ${\mathbf{D}}$ and $J$ as the corresponding
category embedding (up to category equivalence the notion of subcategory
essentially coincides with the notion of fully faithful functor). Let us
formulate the definition of relative right/left adjoint functor for this
particular case in details.

######  Definition 2.11.

Let $F\colon{\mathbf{C}}\to{\mathbf{D}}$ be a functor. A functor
$G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ is called right adjoint for the
functor $F$ relative to the full subcategory
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ iff there exist isomorphisms of the
form (2.6) natural in $X\in{\mathbf{C}}$ and $Z\in{\mathbf{D}}^{\prime}$.
These isomorphisms give a natural isomorphism $\alpha$ between the functors

$\displaystyle{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}^{\prime}\xrightarrow{F^{\mathrm{op}}\times
E_{{\mathbf{D}}^{\prime}}}{\mathbf{D}}^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}}{\mathbf{Set}},$
(2.9)
$\displaystyle{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{D}}^{\prime}\xrightarrow{\mathop{\mathrm{id}}\nolimits_{{\mathbf{C}}^{\mathrm{op}}}\times
G}{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}}{\mathbf{Set}};$
(2.10)

then $(F,G,\alpha)$ is called relative right adjunction and $\alpha$ is called
right adjunction bijection. In this case we say that $F$ has a right adjoint
relative to ${\mathbf{D}}^{\prime}$.

Let $G\colon{\mathbf{D}}\to{\mathbf{C}}$ be a functor. A functor
$F\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ is called left adjoint for the
functor $G$ relative to the full subcategory
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ iff there exist isomorphisms of the
form (2.6) natural in $X\in{\mathbf{C}}^{\prime}$ and $Z\in{\mathbf{D}}$.
These isomorphisms give a natural isomorphism $\alpha$ between the functors

$\displaystyle({\mathbf{C}}^{\prime})^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{F^{\mathrm{op}}\times\mathop{\mathrm{id}}\nolimits_{\mathbf{D}}}{\mathbf{D}}^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}}{\mathbf{Set}},$
(2.11)
$\displaystyle({\mathbf{C}}^{\prime})^{\mathrm{op}}\times{\mathbf{D}}\xrightarrow{E_{({\mathbf{C}}^{\prime})^{\mathrm{op}}}\times
G}{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\xrightarrow{\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}}{\mathbf{Set}};$
(2.12)

then $(F,G,\alpha)$ is called relative left adjunction and $\alpha$ is called
left adjunction bijection. In this case we say that $G$ has a left adjoint
relative to ${\mathbf{C}}^{\prime}$.

Note that $G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ is a right adjoint for
$F\colon{\mathbf{C}}\to{\mathbf{D}}$ relative to
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ iff the opposite functor
$G^{\mathrm{op}}\colon({\mathbf{D}}^{\prime})^{\mathrm{op}}\to{\mathbf{C}}^{\mathrm{op}}$
is a left adjoint for
$F^{\mathrm{op}}\colon{\mathbf{C}}^{\mathrm{op}}\to{\mathbf{D}}^{\mathrm{op}}$
relative to
$({\mathbf{D}}^{\prime})^{\mathrm{op}}\subset{\mathbf{D}}^{\mathrm{op}}$.

######  Warning 2.12.

In contrast to the case of usual adjunction the fact the $F$ is a relative
left adjoint for $G$ does not mean that $G$ is the relative right adjoint for
$F$ and vice versa.

Let ${\mathbf{D}}^{\prime\prime}$ be a full subcategory of
${\mathbf{D}}^{\prime}$. We see from the definition that if
$G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ is a right adjoint for
$F\colon{\mathbf{C}}\to{\mathbf{D}}$ relative to ${\mathbf{D}}^{\prime}$, then
the restriction of $G$ to ${\mathbf{D}}^{\prime\prime}$ is a right adjoint for
$F$ relative to ${\mathbf{D}}^{\prime\prime}$. In particular, if $F$ has a
usual right adjoint (i.e. relative to the whole ${\mathbf{D}}$), then it has a
right adjoint relative to any subcategory
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$. The same we can say about relative
left adjoint functors.

Sometimes a right/left adjoint for a fixed functor does not exists, but there
exists a right/left adjoint relative to some full subcategory. Below we show
how to determine the maximal full subcategory such that a right/left adjoint
exists relative to this subcategory.

A typical example is the forgetful functor
$G\colon{\mathbf{FVect}}\to{\mathbf{Set}}$ from the category of finite-
dimensional vector spaces to the category of sets: the usual left adjoint does
not exists, but we have a left adjoint relative to the subcategory of the
finite sets.

In applications we sometimes have situations, when we are not interested if a
usual right/left adjoint exists, we only need to establish the existence of
the right/left adjoint relative to a fixed full subcategory.

2.3.2. Units and counits for relative adjunctions. Let us generalise some
properties and ‘equivalent definitions’ of the adjoint functors ([MacLane, §
4.1, Theorems 1, 2]) for the case of relative adjoints. In some form they were
formulated in [Ulm] without proof. For completeness we briefly add the proofs
by following ideas of [MacLane] and by applying the corollaries of Yoneda
Lemma.

######  Theorem 2.13.

Consider a functor $F\colon{\mathbf{C}}\to{\mathbf{D}}$ and a full subcategory
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$.

  * $(1).$

Let $G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ be a right adjoint for $F$
relative to ${\mathbf{D}}^{\prime}$. The right adjunction bijections
$\alpha_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(FX,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(X,GZ)$
are inverse to

$\displaystyle\alpha^{-1}(f)=\varepsilon_{Z}\cdot Ff,$ $\displaystyle f\colon
X\to GZ,$ $\displaystyle X\in{\mathbf{C}},$ $\displaystyle
Z\in{\mathbf{D}}^{\prime},$ (2.13)

where $\varepsilon_{Z}:=\alpha^{-1}(\mathop{\mathrm{id}}\nolimits_{GZ})\colon
FGZ\to Z$ are natural in $Z\in{\mathbf{D}}^{\prime}$. The pair
$(GZ,\varepsilon_{Z})$ is a universal morphism from $F$ to
$Z\in{\mathbf{D}}^{\prime}$.

  * $(2).$

Suppose that for any $Z\in{\mathbf{D}}^{\prime}$ there exists a universal
morphism $(X_{Z},\varepsilon_{Z})$ from $F$ to $Z$. Then $F$ has a right
adjoint relative to ${\mathbf{D}}^{\prime}$. This relative right adjoint can
be defined as the unique functor $G\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$
such that $GZ=X_{Z}$ $\;\forall\,Z\in{\mathbf{D}}^{\prime}$ and
$\varepsilon_{Z}$ form a natural transformation from $FG$ to
$E_{{\mathbf{D}}^{\prime}}$. The corresponding right adjunction bijections can
be written via its inverse by the formula (2.13).

Dually, consider a functor $G\colon{\mathbf{D}}\to{\mathbf{C}}$ and a full
subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$.

  * $(1^{\prime}).$

Let $F\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ be a left adjoint for $G$
relative to ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$. The left adjunction
bijections
$\alpha_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(FX,Z)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(X,GZ)$
can be written as

$\displaystyle\alpha(h)=Gh\cdot\eta_{X},$ $\displaystyle h\colon FX\to Z,$
$\displaystyle X\in{\mathbf{C}}^{\prime},$ $\displaystyle Z\in{\mathbf{D}},$
(2.14)

where $\eta_{X}:=\alpha(\mathop{\mathrm{id}}\nolimits_{FX})\colon X\to GFX$
are natural in $X\in{\mathbf{C}}^{\prime}$, and $(FX,\eta_{X})$ is a universal
morphism from $X\in{\mathbf{C}}^{\prime}$ to $G$.

  * $(2^{\prime}).$

Suppose that for any $X\in{\mathbf{C}}^{\prime}$ there exists a universal
morphism $(Z_{X},\eta_{X})$ from $X$ to $G$. Then $G$ has a left adjoint
relative to ${\mathbf{C}}^{\prime}$. This relative left adjoint can be defined
as the unique functor $F\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ such that
$FX=Z_{X}$ $\;\forall\,X\in{\mathbf{C}}^{\prime}$ and $\eta_{X}$ form a
natural transformation from $E_{{\mathbf{C}}^{\prime}}$ to $GF$. The
corresponding left adjunction bijections have the form (2.14).

Proof. The statements $(1)$ and $(2)$ are equivalent to $(1^{\prime})$ and
$(2^{\prime})$ via the categorical duality, so it is sufficient to prove the
latter two.

$(1^{\prime})$: This is a particular case of Corollary 2.5 for $F_{1}=F$ and
$F_{2}=E_{{\mathbf{C}}^{\prime}}$.

$(2^{\prime})$: The functor $F\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ is
constructed as follows. Define it on objects $X\in{\mathbf{C}}^{\prime}$ as
$FX=Z_{X}$. For a morphism $g\colon X\to X^{\prime}$ in
${\mathbf{C}}^{\prime}$ we define $Fg$ as the unique morphism $FX\to
FX^{\prime}$ such that the diagram

(2.19)

is commutative. By construction the functor $F$ is a unique functor
${\mathbf{C}}^{\prime}\to{\mathbf{D}}$ such that $FX=Z_{X}$
$\;\forall\,X\in{\mathbf{C}}^{\prime}$ and the morphisms $\eta_{X}\colon X\to
GFX$ are natural in $X\in{\mathbf{C}}^{\prime}$. Define the maps
$\alpha_{X,Z}\colon\mathop{\mathrm{Hom}}\nolimits(FX,Z)\to\mathop{\mathrm{Hom}}\nolimits(X,GZ)$
by the formula (2.14), which implies that $\alpha_{X,Z}$ are natural in
$Z\in{\mathbf{D}}$. Their naturality in $X\in{\mathbf{C}}^{\prime}$ follows
from the naturality of $\eta$. The bijectivity of the map $\alpha_{X,Z}$
follows from the universality of $(FX,\eta_{X})$, so $F$ is a left adjoint for
$G$ relative to ${\mathbf{C}}^{\prime}$. ∎

The natural transformation $\eta\colon E_{{\mathbf{C}}^{\prime}}\to GF$ with
the components $\eta_{X}:=\alpha_{X,FX}(\mathop{\mathrm{id}}\nolimits_{FX})$
is called unit for the relative left adjunction $(F,G,\alpha)$. Dually, the
natural transformation $\varepsilon\colon FG\to E_{{\mathbf{D}}^{\prime}}$
defined in Theorem 2.13 is called counit for the relative right adjunction
$(F,G,\alpha)$. Note that in general we do not have a unit for a relative
right adjunction, nor a counit for a relative left adjunction.

We see from Theorem 2.13 that the maximal full subcategory
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ such that the functor
$F\colon{\mathbf{C}}\to{\mathbf{D}}$ has a right adjoint relative to
${\mathbf{D}}^{\prime}$ can be found as follows. It consists of all objects
$Z\in{\mathbf{D}}$ such that there is a universal morphism from $F$ to $Z$.
Analogously, the existence of a universal morphism from $X$ to
$G\colon{\mathbf{D}}\to{\mathbf{C}}$ defines the maximal full subcategory
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ such that $G$ has a left adjoint
relative to ${\mathbf{C}}^{\prime}$.

2.3.3. Relative adjunctions with a parameter. For a bifunctor
$F\colon{\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$ we can consider a
family of functors $F_{Y}=F(-,Y)\colon{\mathbf{C}}\to{\mathbf{D}}$
parametrised by objects of the category ${\mathbf{P}}$. If $F_{Y}$ has a right
adjoint $G_{Y}$ for any $Y\in{\mathbf{P}}$, then there is a unique functor
$G\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{D}}\to{\mathbf{C}}$ such that
$G(Y,f)=G_{Y}(f)$ for any morphism $f$ in ${\mathbf{D}}$ and the corresponding
adjunction bijections

$\displaystyle\alpha_{X,Y,Z}\colon\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}F(X,Y),Z\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}\big{(}X,G(Y,Z)\big{)}$
(2.20)

are natural in $X\in{\mathbf{C}}$, $Y\in{\mathbf{P}}$, $Z\in{\mathbf{D}}$ (see
[MacLane, § 4.7, Th. 3]). This fact is immediately generalised to the case of
relative adjunctions.

######  Theorem 2.14.

Let $F\colon{\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$ be a bifunctor such
that the functor $F(-,Y)\colon{\mathbf{C}}\to{\mathbf{D}}$ has a right adjoint
relative to a full subcategory ${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ for
each $Y\in{\mathbf{P}}$, so we have a relative right adjunction
$\big{(}F(-,Y),G_{Y},\alpha_{Y}\big{)}$ for each $Y\in{\mathbf{P}}$, where
$G_{Y}\colon{\mathbf{D}}^{\prime}\to{\mathbf{C}}$ is the relative right
adjoint and $\alpha_{Y}$ is the corresponding right adjunction bijection. Then
there exists a unique functor
$G\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{D}}^{\prime}\to{\mathbf{C}}$
such that $G(Y,-)=G_{Y}$ and the components
$\alpha_{X,Y,Z}=(\alpha_{Y})_{X,Z}$ give the isomorphism of the form (2.20)
natural in $X\in{\mathbf{C}}$, $Y\in{\mathbf{P}}$,
$Z\in{\mathbf{D}}^{\prime}$.

Let $G\colon{\mathbf{D}}\times{\mathbf{P}}\to{\mathbf{C}}$ be a bifunctor such
that each functor $G(-,Y)\colon{\mathbf{D}}\to{\mathbf{C}}$ has a left adjoint
relative to a full subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$, so
we have a relative left adjunction $\big{(}F_{Y},G(-,Y),\alpha_{Y}\big{)}$ for
each $Y\in{\mathbf{P}}$, where
$F_{Y}\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$. Then there exists a unique
functor
$F\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{C}}^{\prime}\to{\mathbf{D}}$
such that $F(Y,-)=F_{Y}$ and the components
$\alpha_{X,Y,Z}=(\alpha_{Y})_{X,Z}$ give the isomorphism of the form

$\displaystyle\alpha_{X,Y,Z}\colon\mathop{\mathrm{Hom}}\nolimits_{\mathbf{D}}\big{(}F(Y,X),Z\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits_{\mathbf{C}}\big{(}X,G(Z,Y)\big{)}$
(2.21)

natural in $X\in{\mathbf{C}}^{\prime}$, $Y\in{\mathbf{P}}$,
$Z\in{\mathbf{D}}$.

Proof. Let us prove the second part (the first part is dual). By having the
functors $F_{Y}=F(Y,-)\colon{\mathbf{C}}^{\prime}\to{\mathbf{D}}$ for
$Y\in{\mathbf{P}}$ we need to construct a bifunctor
$F\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{C}}^{\prime}\to{\mathbf{D}}$
such that (2.21) is natural in $X\in{\mathbf{C}}^{\prime}$, $Y\in{\mathbf{P}}$
and $Z\in{\mathbf{D}}$. This amounts to construction of functors
$\widetilde{F}_{X}=F(-,X)\colon{\mathbf{P}}^{\mathrm{op}}\to{\mathbf{D}}$
satisfying the following conditions: $F_{Y}(X)=\widetilde{F}_{X}(Y)$, the
bijection (2.21) is natural in $Y\in{\mathbf{P}}$ (for each fixed
$X\in{\mathbf{C}}^{\prime}$, $Z\in{\mathbf{D}}$) and the diagram

(2.26)

commutes for any morphisms $f\colon X\to X^{\prime}$ in
${\mathbf{C}}^{\prime}$ and $g\colon Y\to Y^{\prime}$ in ${\mathbf{P}}$ (see
[MacLane, § 2.3, Prop. 1, 2]). For each such $g\colon Y\to Y^{\prime}$
consider the morphism $\beta_{X,Z}(g)$ defined by the commutative diagram

(2.31)

It is natural in $X\in{\mathbf{C}}^{\prime}$ and $Z\in{\mathbf{D}}$, since
$\alpha_{X,Y,Z}$ is so. Hence by virtue of Corollary 2.9 there exist unique
morphisms $F(g,X)\colon F(Y^{\prime},X)\to F(Y,X)$ natural in
$X\in{\mathbf{C}}^{\prime}$ such that $\beta_{X,Z}(g)=F(g,X)^{*}$ for any
$Z\in{\mathbf{D}}$. Note that $\beta_{X,Z}(g_{1}\cdot
g_{2})=\beta_{X,Z}(g_{1})\cdot\beta_{X,Z}(g_{2})$ for any morphisms $g_{1}$
and $g_{2}$ in ${\mathbf{P}}$, so the uniqueness of $F(g,X)$ implies that
$\widetilde{F}_{X}=F(-,X)$ is a functor
${\mathbf{P}}^{\mathrm{op}}\to{\mathbf{D}}$ for each
$X\in{\mathbf{C}}^{\prime}$. We see from the diagram (2.31) that it makes
$\alpha_{X,Y,Z}$ to be natural in $Y\in{\mathbf{P}}$. The uniqueness of the
morphisms implies also that such functor is unique. The naturality of $F(g,X)$
in $X\in{\mathbf{C}}^{\prime}$ means exactly the commutativity of the diagram
(2.26). ∎

######  Definition 2.15.

A right adjunction with a parameter for a bifunctor
$F\colon{\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$ relative to
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ is a triple $(F,G,\alpha)$, where
$G\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{D}}^{\prime}\to{\mathbf{C}}$
is a functor and $\alpha$ is an isomorphism of the form (2.20) natural in
$X\in{\mathbf{C}}$, $Y\in{\mathbf{P}}$, $Z\in{\mathbf{D}}^{\prime}$. A left
adjunction with a parameter for a bifunctor
$G\colon{\mathbf{D}}\times{\mathbf{P}}\to{\mathbf{C}}$ relative to
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ is a triple $(F,G,\alpha)$, where
$F\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{C}}^{\prime}\to{\mathbf{D}}$
is a functor and $\alpha$ is an isomorphism of the form (2.21) natural in
$X\in{\mathbf{C}}^{\prime}$, $Y\in{\mathbf{P}}$, $Z\in{\mathbf{D}}$.

Theorem 2.14 implies the following fact: a right/left adjunction with a
parameter for a bifunctor
$F\colon{\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$ relative to
${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$ exists iff the functor $F(-,Y)$ has
a right/left adjoint relative to ${\mathbf{D}}^{\prime}$ for each object
$Y\in{\mathbf{P}}$.

The counits of the relative right adjunctions
$\big{(}F(-,Y),G(Y,-),\alpha\big{)}$ have components

$\displaystyle\varepsilon_{Y,Z}=\alpha^{-1}(\mathop{\mathrm{id}}\nolimits_{G(Y,Z)})\colon
F\big{(}G(Y,Z),Y\big{)}\to Z.$ (2.32)

These are morphisms in ${\mathbf{D}}$ natural in $Z\in{\mathbf{D}}^{\prime}$.
They are dinatural in $Y\in{\mathbf{P}}$ in the sense of [MacLane, § 9.4].

The units of the relative left adjunctions
$\big{(}F(Y,-),G(-,Y),\alpha\big{)}$ have components

$\displaystyle\eta_{Y,X}=\alpha(\mathop{\mathrm{id}}\nolimits_{F(Y,X)})\colon
X\to G\big{(}F(Y,X),Y\big{)}.$ (2.33)

These are morphisms in ${\mathbf{C}}$ natural in $X\in{\mathbf{C}}^{\prime}$
and dinatural in $Y\in{\mathbf{P}}$.

Note that relative right/left adjunction is a particular case of the relative
right/left adjunction with a parameter. This is the case of the category
${\mathbf{P}}$ which has a unique object and a unique arrow.

2.3.4. Uniqueness of relative adjunctions. If a usual right/left adjoint to a
fixed functor exists, then it is unique up to an isomorphism [MacLane, § 4.1,
Corollary 1]. Moreover, this isomorphism is unique and it relates the
corresponding adjunction bijections. The same still holds for the case of
relative adjunctions. We prove this for more general case: for the relative
adjunctions with a parameter.

######  Theorem 2.16.

If there exist two right adjunctions with a parameter $(F,G,\alpha)$ and
$(F,G^{\prime},\alpha^{\prime})$ for
$F\colon{\mathbf{C}}\times{\mathbf{P}}\to{\mathbf{D}}$ relative to a full
subcategory ${\mathbf{D}}^{\prime}\subset{\mathbf{D}}$, then there exists a
unique natural isomorphism $\gamma\colon
G\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}G^{\prime}$ such that
$(\gamma_{Y,Z})_{*}\cdot\alpha_{X,Y,Z}=\alpha^{\prime}_{X,Y,Z}$ for all
$X\in{\mathbf{C}}$, $Y\in{\mathbf{P}}$, $Z\in{\mathbf{D}}^{\prime}$.

Dually, if there exist two left adjunctions with a parameter $(F,G,\alpha)$
and $(F^{\prime},G,\alpha^{\prime})$ for
$G\colon{\mathbf{D}}\times{\mathbf{P}}\to{\mathbf{C}}$ relative to a full
subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$, then there exists a
unique natural isomorphism $\gamma\colon
F\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F^{\prime}$ such that
$\alpha_{X,Y,Z}\cdot(\gamma_{Y,X})^{*}=\alpha^{\prime}_{X,Y,Z}$ for all
$X\in{\mathbf{C}}^{\prime}$, $Y\in{\mathbf{P}}$, $Z\in{\mathbf{D}}$.

Proof. As above we prove the ‘left’ version. Consider the compositions

$\displaystyle\mathop{\mathrm{Hom}}\nolimits\big{(}F^{\prime}(Y,X),Z\big{)}\xrightarrow{\alpha^{\prime}_{X,Y,Z}}\mathop{\mathrm{Hom}}\nolimits\big{(}X,G(Z,Y)\big{)}\xrightarrow{\alpha_{X,Y,Z}^{-1}}\mathop{\mathrm{Hom}}\nolimits\big{(}F(Y,X),Z\big{)}.$
(2.34)

These are bijections
$\beta_{Y,X,Z}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}F^{\prime}(Y,X),Z\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}F(Y,X),Z\big{)}$
natural in $X\in{\mathbf{C}}^{\prime}$, $Y\in{\mathbf{P}}$ and
$Z\in{\mathbf{D}}$. By virtue of Corollary 2.10 they have the form
$\beta_{Y,X,Z}=\gamma_{Y,X}^{*}$ for a unique natural isomorphism
$\gamma\colon F\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F^{\prime}$.
∎

######  Remark 2.17.

The uniqueness property of relative adjunctions (with a parameter) can be
rewritten in terms of counits (2.32) and units (2.33) respectively. Let
$\eta_{Y,X}=\alpha(\mathop{\mathrm{id}}\nolimits_{F(Y,X)})$ and
$\eta_{Y,X}=\alpha^{\prime}(\mathop{\mathrm{id}}\nolimits_{F^{\prime}(Y,X)})$
be units for the relative left adjunctions with a parameter $(F,G,\alpha)$ and
$(F^{\prime},G,\alpha^{\prime})$ respectively. The natural isomorphism
$\gamma\colon F\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F^{\prime}$
from Theorem 2.16 relates these units by the formula
$\eta^{\prime}_{Y,X}=G(\gamma_{Y,X},\mathop{\mathrm{id}}\nolimits_{Y})\cdot\eta_{Y,X}$.
The existence and uniqueness of such isomorphisms $\gamma_{Y,X}\colon
F(Y,X)\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}F^{\prime}(Y,X)$
follows from the fact that a universal morphism from $X$ to $G(-,Y)$ is unique
up to a unique isomorphism (see Remark 2.8). Analogously, the counits
$\varepsilon_{Y,Z}=\alpha^{-1}(\mathop{\mathrm{id}}\nolimits_{G(Y,Z)})$ and
$\varepsilon^{\prime}_{Y,Z}=(\alpha^{\prime})^{-1}(\mathop{\mathrm{id}}\nolimits_{G^{\prime}(Y,Z)})$
are related as $\varepsilon^{\prime}_{Y,Z}\cdot
F(\gamma_{Z,Y},\mathop{\mathrm{id}}\nolimits_{Y})=\varepsilon_{Y,Z}$ for
unique $\gamma_{Z,Y}$.

###  2.4 Relatively closed monoidal categories

Representations can be defined in a closed monoidal category. But sometimes we
need to consider more general situation. The notion of adjunction (with a
parameter) relative to a full subcategory allows us to generalise closed
monoidal category to the case sufficient for our purposes.

2.4.1. Generalisations of closed and coclosed monoidal categories. Let
${\mathbf{C}}=({\mathbf{C}},\otimes)$ be a symmetric monoidal category with a
monoidal product
$-\otimes-\colon{\mathbf{C}}\times{\mathbf{C}}\to{\mathbf{C}}$. As in [S21] we
suppose that the monoidal categories are strict. We denote the unit object of
${\mathbf{C}}$ by $I_{\mathbf{C}}$. Recall that ${\mathbf{C}}$ is called
closed monoidal category iff the functor $-\otimes
V\colon{\mathbf{C}}\to{\mathbf{C}}$ has a right adjoint for each
$V\in{\mathbf{C}}$. This right adjoint is denoted as
$h\mapsto\mathop{\mathrm{\bf hom}}\nolimits(V,h)$ and gives a bifunctor
$\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{C}}$
called internal hom-functor. Since ${\mathbf{C}}$ is symmetric, the right
adjoint functors for $-\otimes V$ and $V\otimes-$ coincide with each other.
Dually, ${\mathbf{C}}$ is called coclosed monoidal category iff $-\otimes V$
has a left adjoint for each $V\in{\mathbf{C}}$. In this case we obtain
internal cohom-functor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{C}}$.

For purposes of Quantum Representation Theory we considered more general
situation in [S21], when the functors $-\otimes V$ have right/left adjoints
for the objects $V$ of a full subcategory ${\mathbf{P}}\subset{\mathbf{C}}$
(we do not always suppose that this subcategory is monoidal). For this
situation we introduced representations/corepresentations of a monoid/comonoid
in ${\mathbf{C}}$ on an object $V\in{\mathbf{P}}$. However the condition for
these functors to have adjoints is too strong. Here we consider a weaker
condition by requiring the existence of right/left adjoints relative to
${\mathbf{P}}\subset{\mathbf{C}}$.

######  Definition 2.18.

The symmetric monoidal category ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is
called closed/coclosed relative to a full subcategory
${\mathbf{P}}\subset{\mathbf{C}}$ or relatively closed/coclosed with a
parametrising subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ iff the functor
$-\otimes V\colon{\mathbf{C}}\to{\mathbf{C}}$ has a right/left adjoint
relative to ${\mathbf{P}}\subset{\mathbf{C}}$ for each object
$V\in{\mathbf{P}}$.

The following facts are proved by restricting the corresponding relative
adjoint functors.

######  Proposition 2.19.

Let ${\mathbf{C}}=({\mathbf{C}},\otimes)$ be a symmetric monoidal category and
${\mathbf{P}}\subset{\mathbf{C}}$ be its full subcategory.

  * (1).

If the monoidal category ${\mathbf{C}}$ is closed/coclosed relative to
${\mathbf{P}}$, then it is closed/coclosed relative to any full subcategory
${\mathbf{P}}^{\prime}\subset{\mathbf{P}}$.

  * (2).

Let ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ be a full subcategory such that
${\mathbf{P}}\subset{\mathbf{C}}^{\prime}$. If the functor $-\otimes
V\colon{\mathbf{C}}\to{\mathbf{C}}$ has a right/left adjoint relative to
${\mathbf{C}}^{\prime}$ for each $V\in{\mathbf{P}}$, then
$({\mathbf{C}},\otimes)$ is closed/coclosed relative to ${\mathbf{P}}$.

  * (3).

In particular, if the functor $-\otimes V\colon{\mathbf{C}}\to{\mathbf{C}}$
has a right/left adjoint for each $V\in{\mathbf{P}}$, then the monoidal
category ${\mathbf{C}}$ is closed/coclosed relative to ${\mathbf{P}}$.

The condition supposed in the point (3) of Prop. 2.19 is often fulfilled,
however this stronger condition is not needed to construct the representation
theory. The subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ is usually chosen as
bigger as possible such that the functor $-\otimes V$ has a right/left
(relative) adjoint for all $V\in{\mathbf{P}}$. In some cases it is easier to
check that these functors have adjoints relative to ${\mathbf{P}}$ than
relative to the whole ${\mathbf{C}}$.

The simplest non-trivial example is the category of vector spaces
${\mathbf{C}}={\mathbf{Vect}}$ with the standard tensor product $\otimes$,
where the parametrising subcategory is ${\mathbf{P}}={\mathbf{FVect}}$. The
functor $-\otimes V\colon{\mathbf{Vect}}\to{\mathbf{Vect}}$ has a left adjoint
for any $V\in{\mathbf{FVect}}$ (see [S21, § 3.1.11]), so due to the point (3)
of Prop. 2.19 the symmetric monoidal category $({\mathbf{Vect}},\otimes)$ is
coclosed relative to ${\mathbf{FVect}}$.

2.4.2. Internal hom. Due to Theorem 2.14 (for the case
${\mathbf{D}}={\mathbf{C}}$) a symmetric monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes)$ is relatively closed with the
parametrising subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ iff there exists a
bifunctor $\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$
with the bijections

$\displaystyle\theta=\theta_{X,V,W}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(X\otimes
V,W)$ (2.35)

natural in $X\in{\mathbf{C}}$, $V,W\in{\mathbf{P}}$. Or, equivalently, iff
there exists a right adjunction with a parameter
$(-\otimes-,\mathop{\mathrm{\bf hom}}\nolimits,\theta^{-1})$ relative to the
full subcategory ${\mathbf{D}}^{\prime}={\mathbf{P}}$. The bifunctor
$\mathop{\mathrm{\bf hom}}\nolimits$ is called (relative) internal hom-
functor. The values $\mathop{\mathrm{\bf hom}}\nolimits(V,W)$ on the objects
$V,W\in{\mathbf{P}}$ are called internal hom-objects.

One can consider the bifunctor $\mathop{\mathrm{\bf hom}}\nolimits$ with the
natural isomorphism $\theta$ as an additional structure on the symmetric
monoidal category ${\mathbf{C}}$. However, Theorem 2.16 implies that the
relative right adjunction with a parameter
$\big{(}-\otimes-,\mathop{\mathrm{\bf hom}}\nolimits,\theta^{-1}\big{)}$ is
unique up to a unique natural isomorphism. This means that the choice of the
certain pair $(\mathop{\mathrm{\bf hom}}\nolimits,\theta)$ is not essential,
all the choices are equivalent.

Dually, a symmetric monoidal category ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is
relatively coclosed with the parametrising subcategory
${\mathbf{P}}\subset{\mathbf{C}}$ iff there exists a bifunctor
$\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$,
called (relative) internal cohom-functor, with the bijections

$\displaystyle\vartheta=\vartheta_{W,V,Z}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(V,W),Z\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(W,Z\otimes
V)$ (2.36)

natural in $V,W\in{\mathbf{P}}$, $Z\in{\mathbf{C}}$. This means that
$(-\otimes-,\mathop{\mathrm{\bf cohom}}\nolimits,\vartheta)$ is a left
adjunction with a parameter relative to the full subcategory
${\mathbf{C}}^{\prime}={\mathbf{P}}$; it is also unique up to an isomorphism.
The values $\mathop{\mathrm{\bf cohom}}\nolimits(V,W)$ on the objects
$V,W\in{\mathbf{P}}$ are called internal cohom-objects.

######  Proposition 2.20.

Let ${\mathbf{C}}=({\mathbf{C}},\otimes)$ be a symmetric monoidal category and
${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ be its full monoidal subcategory.
Consider also a full subcategory ${\mathbf{P}}\subset{\mathbf{C}}^{\prime}$.

  * (1).

If ${\mathbf{C}}$ is closed relative to ${\mathbf{P}}$ and
$\mathop{\mathrm{\bf hom}}\nolimits(V,W)\in{\mathbf{C}}^{\prime}$
$\;\forall\,V,W\in{\mathbf{P}}$, then $({\mathbf{C}}^{\prime},\otimes)$ is
also closed relative to ${\mathbf{P}}$. The internal hom-functor
$\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$
equals to the composition of $\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}^{\prime}$
with the embedding ${\mathbf{C}}^{\prime}\hookrightarrow{\mathbf{C}}$.

  * (2).

If ${\mathbf{C}}$ is coclosed relative to ${\mathbf{P}}$ and
$\mathop{\mathrm{\bf cohom}}\nolimits(V,W)\in{\mathbf{C}}^{\prime}$
$\;\forall\,V,W\in{\mathbf{P}}$, then $({\mathbf{C}}^{\prime},\otimes)$ is
also coclosed relative to ${\mathbf{P}}$. The bifunctor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$
equals to the composition of $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}^{\prime}$
with ${\mathbf{C}}^{\prime}\hookrightarrow{\mathbf{C}}$.

Proof. The condition of the point (1) imply that the bifunctor
$\mathop{\mathrm{\bf
hom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$
has the form $\mathop{\mathrm{\bf
hom}}\nolimits=E_{{\mathbf{C}}^{\prime}}\cdot G$ for some
$G\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}^{\prime}$.
For any $V\in{\mathbf{P}}$ the functor
$G(V,-)\colon{\mathbf{P}}\to{\mathbf{C}}^{\prime}$ is a right adjoint for
$-\otimes V\colon{\mathbf{C}}^{\prime}\to{\mathbf{C}}^{\prime}$ relative to
${\mathbf{P}}$. ∎

2.4.3. Composition morphisms. Let ${\mathbf{C}}=({\mathbf{C}},\otimes)$ be a
relatively closed monoidal category with the parametrising subcategory
${\mathbf{P}}$. Counits (2.32) for $(-\otimes-,\mathop{\mathrm{\bf
hom}}\nolimits,\theta^{-1})$ are morphisms

$\displaystyle\mathop{\mathrm{ev}}\nolimits_{V,W}=\theta(\mathop{\mathrm{id}}\nolimits_{\mathop{\mathrm{\bf
hom}}\nolimits(V,W)})\colon\mathop{\mathrm{\bf hom}}\nolimits(V,W)\otimes V\to
W,$ $\displaystyle V,W\in{\mathbf{P}}.$ (2.37)

called evaluations. The formula (2.13) takes the form

$\displaystyle\theta_{X,V,W}(f)=\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot(f\otimes\mathop{\mathrm{id}}\nolimits_{V}),$
$\displaystyle X\in{\mathbf{C}},\;\;V,W\in{\mathbf{P}},\quad f\colon
X\to\mathop{\mathrm{\bf hom}}\nolimits(V,W).$ (2.38)

This implies the formula

$\displaystyle\theta_{\widetilde{X},V,W}(f\cdot
g)=\theta_{X,V,W}(f)\cdot(g\otimes\mathop{\mathrm{id}}\nolimits_{V}),$ (2.39)

where $X,\widetilde{X}\in{\mathbf{C}}$, $V,W\in{\mathbf{P}}$ and $f\colon
X\to\mathop{\mathrm{\bf hom}}\nolimits(V,W)$, $g\colon\widetilde{X}\to X$ are
morphisms in ${\mathbf{C}}$ (see [S21, Prop. 3.1]).

For any closed monoidal category one can define internal compositions (see
[Bor2, § 6.1]). Let us generalise them for the relative case. Consider the
morphisms

$\displaystyle\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf hom}}\nolimits(U,V)\otimes
U\xrightarrow{\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U,V}}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\otimes
V\xrightarrow{\mathop{\mathrm{ev}}\nolimits_{V,W}}W,$ (2.40)

where $U,V,W\in{\mathbf{P}}$. By applying the map $\theta^{-1}_{X,U,W}$, where
$X=\mathop{\mathrm{\bf hom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
hom}}\nolimits(U,V)$, to the morphism (2.40) we obtain

$\displaystyle c_{U,V,W}\colon\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
hom}}\nolimits(U,V)\to\mathop{\mathrm{\bf hom}}\nolimits(U,W).$ (2.41)

It is called (internal) composition morphism. It is the unique morphism of the
form (2.41) such that the diagram

(2.46)

is commutative.

In the case of relatively coclosed monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes)$ with parametrising subcategory
${\mathbf{P}}$ we have units (2.33) of the form

$\displaystyle\mathop{\mathrm{coev}}\nolimits_{V,W}=\vartheta(\mathop{\mathrm{id}}\nolimits_{\mathop{\mathrm{\bf
hom}}\nolimits(V,W)})\colon W\to\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes V,$ $\displaystyle V,W\in{\mathbf{P}}.$ (2.47)

They are called coevaluations. We get the formula

$\displaystyle\vartheta(f)=(f\otimes\mathop{\mathrm{id}}\nolimits_{V})\cdot\mathop{\mathrm{coev}}\nolimits_{V,W},$
$\displaystyle f\colon\mathop{\mathrm{\bf cohom}}\nolimits(V,W)\to Z.$ (2.48)

The (internal) cocomposition morphism is

$\displaystyle
d_{W,V,U}=\vartheta^{-1}\big{(}(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{coev}}\nolimits_{U,V})\cdot\mathop{\mathrm{coev}}\nolimits_{V,W}\big{)}\colon\mathop{\mathrm{\bf
cohom}}\nolimits(U,W)\to\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf cohom}}\nolimits(U,V),$ (2.49)

where $U,V,W\in{\mathbf{P}}$. This is a unique morphism making the following
diagram commutative:

(2.54)

######  Proposition 2.21.

The composition (2.41) is associative: the diagram

(2.59)

commutes for any $T,U,V,W\in{\mathbf{P}}$. Dually, cocomposition (2.49) is
coassociative in the sense of commutativity of the diagram

Proof. By drawing the commutative diagram (2.46) for each composition morphism
in (2.59) one yields

$\mathop{\mathrm{ev}}\nolimits_{T,W}\cdot(c_{T,U,W}\otimes\mathop{\mathrm{id}}\nolimits_{T})\cdot(c_{U,V,W}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits_{T})=\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U,V})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{T,U})=\\\
\mathop{\mathrm{ev}}\nolimits_{T,W}\cdot(c_{T,V,W}\otimes\mathop{\mathrm{id}}\nolimits_{T})\cdot(\mathop{\mathrm{id}}\nolimits\otimes
c_{T,U,V}\otimes\mathop{\mathrm{id}}\nolimits_{T}).$

This implies commutativity of the diagram (2.59) due to the formula (2.38) and
bijectivity of $\theta$ (or, equivalently, due to the universality of
$\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(T,W),\mathop{\mathrm{ev}}\nolimits_{T,W}\big{)}$). The
coassociativity of the cocomposition then follows from the duality principle.
∎

2.4.4. Internal end. Consider the bijection (2.35) with the unit object
$X=I_{\mathbf{C}}$ and $W=V$. Denote the image of the identification
isomorphism $I\otimes V=V$ under $\theta^{-1}$ by

$\displaystyle u_{V}:=\theta^{-1}(\mathop{\mathrm{id}}\nolimits_{V})\colon
I_{\mathbf{C}}\to\mathop{\mathrm{\bf hom}}\nolimits(V,V),$ $\displaystyle
V\in{\mathbf{P}}.$ (2.60)

This is a unique morphism making the following diagram commute:

(2.65)

######  Lemma 2.22.

For any $V,W\in{\mathbf{P}}$ we have the commutative diagrams

(2.70)

Proof. The commutativity of these diagrams can be shown by means of the
formula (2.39). Let us consider the right diagram (2.70). By applying the
bijection $\theta$ one yields

$\theta\big{(}c_{V,V,W}\cdot(\mathop{\mathrm{id}}\nolimits\otimes
u_{V})\big{)}=\theta(c_{V,V,W})\cdot(\mathop{\mathrm{id}}\nolimits\otimes
u_{V}\otimes\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{V,V})\cdot(\mathop{\mathrm{id}}\nolimits\otimes
u_{V}\otimes\mathop{\mathrm{id}}\nolimits)=\\\
\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot\big{(}\mathop{\mathrm{id}}\nolimits\otimes\theta(u_{V})\big{)}=\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot\big{(}\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits\big{)}=\theta(\mathop{\mathrm{id}}\nolimits).$

The commutativity of the left diagram is proved similarly. ∎

Denote $\mathop{\mathrm{\bf end}}\nolimits(V)=\mathop{\mathrm{\bf
hom}}\nolimits(V,V)$. Proposition 2.21 for the case $U=V=W$ and Lemma 2.22 for
$V=W$ imply that the morphisms $c_{V}:=c_{V,V,V}\colon\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V)\to\mathop{\mathrm{\bf end}}\nolimits(V)$ and $u_{V}\colon
I_{\mathbf{C}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$ give a structure of
monoid in ${\mathbf{C}}=({\mathbf{C}},\otimes)$ on the object
$\mathop{\mathrm{\bf end}}\nolimits(V)\in{\mathbf{C}}$.

Dually, by applying the inverse of the bijection (2.36), where $W=V$ and
$Z=I_{\mathbf{C}}$, to the identification isomorphism $V=I\otimes V$ we obtain
the morphism

$\displaystyle
v_{V}:=\vartheta^{-1}(\mathop{\mathrm{id}}\nolimits_{V})\colon\mathop{\mathrm{\bf
cohom}}\nolimits(V,V)\to I_{\mathbf{C}},$ $\displaystyle V\in{\mathbf{P}}.$
(2.71)

This is a unique morphism making the following diagram commute:

(2.76)

A symmetric monoidal category ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is closed
relative to ${\mathbf{P}}$ iff its opposite category
${\mathbf{C}}^{\mathrm{op}}=({\mathbf{C}}^{\mathrm{op}},\otimes)$ is coclosed
relative to ${\mathbf{P}}^{\mathrm{op}}$. The hom-object $\mathop{\mathrm{\bf
hom}}\nolimits(V,W)$ for ${\mathbf{C}}$ is the cohom-object
$\mathop{\mathrm{\bf cohom}}\nolimits(V,W)$ for ${\mathbf{C}}^{\mathrm{op}}$.
The morphisms $\theta_{X,V,W}$ and $\mathop{\mathrm{ev}}\nolimits_{V,W}$ for
${\mathbf{C}}$ are $\vartheta_{W,V,X}$ and
$\mathop{\mathrm{coev}}\nolimits_{V,W}$ for ${\mathbf{C}}^{\mathrm{op}}$. The
composition $c_{U,V,W}$ and morphism $u_{V}$ for ${\mathbf{C}}$ are the
cocomposition $d_{W,V,U}$ and morphism $v_{V}$ for
${\mathbf{C}}^{\mathrm{op}}$. By taking into account this duality and argument
mentioned above one can state the following.

######  Proposition 2.23.

If ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is relatively closed monoidal
category with parametrising subcategory ${\mathbf{P}}\subset{\mathbf{C}}$,
then for each $V\in{\mathbf{P}}$ we have the monoid $\mathop{\mathrm{\bf
end}}\nolimits(V)=\big{(}\mathop{\mathrm{\bf
end}}\nolimits(V),c_{V},u_{V}\big{)}$ in $({\mathbf{C}},\otimes)$, where
$c_{V}:=c_{V,V,V}$ and $u_{V}$ is (2.60).

Dually, if ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is relatively coclosed
monoidal category with parametrising subcategory
${\mathbf{P}}\subset{\mathbf{C}}$, then for each $V\in{\mathbf{P}}$ the
morphisms $d_{V}:=d_{V,V,V}$ and (2.71) equip the object $\mathop{\mathrm{\bf
coend}}\nolimits(V):=\mathop{\mathrm{\bf cohom}}\nolimits(V,V)$ with a
structure of comonoid $\mathop{\mathrm{\bf
coend}}\nolimits(V)=\big{(}\mathop{\mathrm{\bf
coend}}\nolimits(V),d_{V},v_{V}\big{)}$ in $({\mathbf{C}},\otimes)$.

The diagrams (2.46) for $U=V=W$ and (2.65) imply that
$\mathop{\mathrm{ev}}\nolimits_{V,V}\colon\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes V\to V$ is a (left) action of the monoid
$\mathop{\mathrm{\bf end}}\nolimits(V)$ on $V$. It also follows from the
diagrams (2.54) for $U=V=W$ and (2.76) that
$\mathop{\mathrm{coev}}\nolimits_{V,V}\colon V\to\mathop{\mathrm{\bf
coend}}\nolimits(V)\otimes V$ is a (left) coaction of the comonoid
$\mathop{\mathrm{\bf coend}}\nolimits(V)$ on $V$.

2.4.5. The hom-functor on morphisms. The hom-functor as well as cohom-functor
is a bifunctor contravariant in the first and covariant in the second
argument, i.e. a functor
${\mathbf{C}}^{\mathrm{op}}\times{\mathbf{C}}\to{\mathbf{C}}$. Let us
calculate their values on morphisms in terms of evaluation and coevaluation.

######  Proposition 2.24.

Let $f\colon V^{\prime}\to V$ and $g\colon W\to W^{\prime}$ be morphisms in
${\mathbf{P}}$. In the relatively closed case we have

$\displaystyle\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f,g)\big{)}=g\cdot\mathop{\mathrm{ev}}\nolimits_{V,W}\cdot(\mathop{\mathrm{id}}\nolimits_{\mathop{\mathrm{\bf
hom}}\nolimits(V,W)}\otimes f).$ (2.77)

For the relatively coclosed case the following formula holds:

$\displaystyle\vartheta\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(f,g)\big{)}=(\mathop{\mathrm{id}}\nolimits_{\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime})}\otimes
f)\cdot\mathop{\mathrm{coev}}\nolimits_{V^{\prime},W^{\prime}}\cdot g.$ (2.78)

Proof. Repeats the proof of [S21, Prop. 3.4]. ∎

2.4.6. The case $V=I_{\mathbf{C}}$. Suppose that
$I=I_{\mathbf{C}}\in{\mathbf{P}}$. Then we can define the hom-objects
$\mathop{\mathrm{\bf hom}}\nolimits(I,W)\in{\mathbf{C}}$ and the monoid
$\mathop{\mathrm{\bf
end}}\nolimits(I)\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$ (in the
relatively closed case) or the cohom-objects $\mathop{\mathrm{\bf
cohom}}\nolimits(I,W)\in{\mathbf{C}}$ and the comonoid $\mathop{\mathrm{\bf
coend}}\nolimits(I)\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$ (in the
relatively coclosed case). We can describe them as follows (see [Bor2, Prop.
6.1.8] for the usual closed case).

######  Proposition 2.25.

The object $\mathop{\mathrm{\bf hom}}\nolimits(I,W)$ is naturally isomorphic
to $W$ via the isomorphism

$\displaystyle\mathop{\mathrm{\bf hom}}\nolimits(I,W)=\mathop{\mathrm{\bf
hom}}\nolimits(I,W)\otimes
I\xrightarrow{\mathop{\mathrm{ev}}\nolimits_{I,W}}W.$ (2.79)

In particular, $\mathop{\mathrm{ev}}\nolimits_{I,I}$ gives the identification
of objects $\mathop{\mathrm{\bf end}}\nolimits(I)=I$. As a monoid it coincides
with $\mathop{\mathrm{\bf
end}}\nolimits(I)=(I,\mathop{\mathrm{id}}\nolimits_{I},\mathop{\mathrm{id}}\nolimits_{I})$.
The object $\mathop{\mathrm{\bf cohom}}\nolimits(I,W)$ is naturally isomorphic
to $W$ via

$\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits(I,W)=W\xrightarrow{\mathop{\mathrm{coev}}\nolimits_{I,W}}\mathop{\mathrm{\bf
cohom}}\nolimits(I,W)\otimes I.$ (2.80)

In particular, $\mathop{\mathrm{coev}}\nolimits_{I,I}$ gives the
identification $\mathop{\mathrm{\bf coend}}\nolimits(I)=I$. As a comonoid it
has the form $\mathop{\mathrm{\bf
coend}}\nolimits(I)=(I,\mathop{\mathrm{id}}\nolimits_{I},\mathop{\mathrm{id}}\nolimits_{I})$.

Proof. We prove it for the relatively coclosed case. The morphisms (2.80) are
natural in $W\in{\mathbf{P}}$, so we need to prove that they are isomorphisms.
Due to the formula (2.48) the natural isomorphism

$\displaystyle\beta_{W,Z}=\vartheta_{W,I,Z}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(I,W),Z\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits(W,Z\otimes
I)=\mathop{\mathrm{Hom}}\nolimits(W,Z)$ (2.81)

has the form $\beta_{W,Z}=(\mathop{\mathrm{coev}}\nolimits_{I,W})^{*}$. Then,
Corollary 2.7 implies that the coevaluations
$\mathop{\mathrm{coev}}\nolimits_{I,W}$ are all isomorphisms. To show that the
last sentence of Prop. 2.25 is also valid we note that all the arrows in the
diagrams (2.54) and (2.76) for $U=V=W=I$ are identifications
$\mathop{\mathrm{id}}\nolimits_{I}$. This implies that
$d_{I}=\mathop{\mathrm{id}}\nolimits_{I}$ and
$v_{I}=\mathop{\mathrm{id}}\nolimits_{I}$. ∎

2.4.7. The case of coreflective relatively coclosed monoidal subcategory.
Recall that a subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ is called
coreflective iff the inclusion
$E_{{\mathbf{C}}^{\prime}}\colon{\mathbf{C}}^{\prime}\hookrightarrow{\mathbf{C}}$
has a right adjoint $G\colon{\mathbf{C}}\to{\mathbf{C}}^{\prime}$ (see
[MacLane, § 4.3]). Due to Theorem 2.13 this condition means exactly that for
any $X\in{\mathbf{C}}$ there exists a universal arrow $(GX,\varepsilon_{X})$
from $E_{{\mathbf{C}}^{\prime}}$ to $X$, where the universality means that for
any object $X^{\prime}\in{\mathbf{C}}^{\prime}$ and morphism $f\colon
X^{\prime}\to X$ in ${\mathbf{C}}$ there is a unique $h\colon X^{\prime}\to
GX$ such that the diagram

(2.86)

is commutative. We choose the adjunction such that $GX^{\prime}=X^{\prime}$,
$\eta_{X^{\prime}}=\varepsilon_{X^{\prime}}=\mathop{\mathrm{id}}\nolimits_{X^{\prime}}$
$\;\forall\,X^{\prime}\in{\mathbf{P}}$. Since $\varepsilon_{X}$ is natural in
$X$, this choice implies that the morphisms in (2.86) are related as $h=Gf$.

Suppose that the subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ is
monoidal with respect to $\otimes$. In particular, we have $GX\otimes
GY\in{\mathbf{C}}^{\prime}$ for any objects $X,Y\in{\mathbf{C}}$. The
commutative diagram (2.86) for $f=\varepsilon_{X}\otimes\varepsilon_{Y}\colon
GX\otimes GY\to X\otimes Y$ takes the form

(2.91)

where $\phi_{X,Y}=G(\varepsilon_{X}\otimes\varepsilon_{Y})$ (this is lax
monoidal structure morphisms of the right adjoint $G$). The following fact
will be needed in Subsection 4.2.

######  Theorem 2.26.

Let ${\mathbf{C}}^{\prime}$ be a coreflective monoidal subcategory of
$({\mathbf{C}},\otimes)$ and let $({\mathbf{C}}^{\prime},\otimes)$ be coclosed
relative to a full subcategory ${\mathbf{P}}\subset{\mathbf{C}}^{\prime}$.
Suppose that $\phi_{X,W}\colon GX\otimes W\to G(X\otimes W)$ is an isomorphism
for any $X\in{\mathbf{C}}$ and $W\in{\mathbf{P}}$. Then
$({\mathbf{C}},\otimes)$ is also coclosed relative to ${\mathbf{P}}$. The
internal cohom-functor $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}$
is the composition of $\mathop{\mathrm{\bf
cohom}}\nolimits\colon{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\to{\mathbf{C}}^{\prime}$
with the embedding
$E_{{\mathbf{C}}^{\prime}}\colon{\mathbf{C}}^{\prime}\hookrightarrow{\mathbf{C}}$.

Proof. Let $V,W\in{\mathbf{P}}$ and $X\in{\mathbf{C}}$. The isomorphism
$GX\otimes W\cong G(X\otimes W)$, universality of $\big{(}G(X\otimes
W),\varepsilon_{X\otimes W}\big{)}$ and commutativity of the diagram (2.91)
for $Y=W$ imply that for any $f\colon V\to X\otimes W$ there exists a unique
$f^{\prime}\colon V\to GX\otimes W$ such that
$(\varepsilon_{X}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot f^{\prime}=f$.
Since $GX\in{\mathbf{C}}^{\prime}$ and ${\mathbf{C}}^{\prime}$ is coclosed
relative to ${\mathbf{P}}$, there is a unique
$h^{\prime}\colon\mathop{\mathrm{\bf cohom}}\nolimits(W,V)\to GX$ such that
$(h^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot\mathop{\mathrm{coev}}\nolimits_{W,V}=f^{\prime}$,
so we have the commutative diagrams

where $h=\varepsilon_{X}\cdot h^{\prime}$. Thus we proved that for any
$f\colon V\to X\otimes W$ there exists a morphism $h\colon\mathop{\mathrm{\bf
cohom}}\nolimits(W,V)\to X$ making the second diagram commute. Now we only
need to prove that it is unique. Let $g\colon\mathop{\mathrm{\bf
cohom}}\nolimits(W,V)\to X$ be such that
$(g\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot\mathop{\mathrm{coev}}\nolimits_{W,V}=f$.
Since $\mathop{\mathrm{\bf cohom}}\nolimits(W,V)\in{\mathbf{C}}^{\prime}$, it
factors as $g=\varepsilon_{X}\cdot g^{\prime}$ for some
$g^{\prime}\colon\mathop{\mathrm{\bf cohom}}\nolimits(W,V)\to GX$, so we
obtain
$(\varepsilon_{X}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot(g^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot\mathop{\mathrm{coev}}\nolimits_{W,V}=f=(\varepsilon_{X}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot
f^{\prime}$. Due to the uniqueness of $f^{\prime}$ we derive
$(g^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot\mathop{\mathrm{coev}}\nolimits_{W,V}=f^{\prime}=(h^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{W})\cdot\mathop{\mathrm{coev}}\nolimits_{W,V}$.
By taking into account the uniqueness of $h^{\prime}$ we get
$g^{\prime}=h^{\prime}$ and, hence, $g=h$. ∎

######  Remark 2.27.

By the duality principle one can state an analogous theorem for a reflective
monoidal subcategory
$({\mathbf{C}}^{\prime},\otimes)\subset({\mathbf{C}},\otimes)$ closed relative
to ${\mathbf{P}}$. In this case the inclusion functor
$E_{{\mathbf{C}}^{\prime}}\colon{\mathbf{P}}\hookrightarrow{\mathbf{C}}$ has a
left adjoint functor $F\colon{\mathbf{C}}\to{\mathbf{P}}$, which is colax
monoidal. If $\phi_{X,W}\colon F(X\otimes W)\to FX\otimes FW$ are isomorphisms
for all $X\in{\mathbf{C}}$ and $W\in{\mathbf{P}}$, then
$({\mathbf{C}},\otimes)$ is also closed relative to ${\mathbf{P}}$.

######  Remark 2.28.

Note that not every symmetric monoidal category is (co)closed relative to a
(co)closed monoidal subcategory. Consider a counterexample. Let ${\mathbf{C}}$
be the partial order of subsets $X\subset{\mathbb{R}}$ with respect to
inclusion. Define the monoidal product as union: $X\otimes Y=X\cup Y$. The
subcategory ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$ consisting of open
subsets is monoidal. It has a monoidal subcategory ${\mathbf{P}}$ of the
intervals $(0,x)$, where $x\in{\mathbb{R}}_{\geqslant 0}$. By identifying
$(0,x)$ with $x$ we can interpret ${\mathbf{P}}$ as the standard normal order
on ${\mathbb{R}}_{\geqslant 0}$ with the monoidal product $x\otimes
y=\max(x,y)$. The monoidal category $({\mathbf{P}},\otimes)$ is coclosed with
the cohom-functor

$\displaystyle\mathop{\mathrm{\bf
cohom}}\nolimits(x,y)=\begin{cases}0,&x\geqslant y;\\\ y,&x<y.\end{cases}$
(2.92)

Nevertheless, the monoidal category $({\mathbf{C}}^{\prime},\otimes)$ is not
coclosed relative to ${\mathbf{P}}$. Theorem 2.26 is not applicable here,
since the subcategory ${\mathbf{P}}\subset{\mathbf{C}}^{\prime}$ is not
coreflective. The whole category ${\mathbf{C}}$ is coclosed relative to
${\mathbf{P}}$ but with another cohom-functor

$\displaystyle\mathop{\mathrm{\bf cohom}}\nolimits(x,y)=\\{a>0\mid x\leqslant
a<y\\}=\begin{cases}\emptyset,&x\geqslant y;\\\ (0,y),&x=0;\\\
[x,y),&0<x<y.\end{cases}$ (2.93)

This gives two counterexamples: for ${\mathbf{C}}^{\prime}\subset{\mathbf{C}}$
and ${\mathbf{P}}\subset{\mathbf{C}}$. The former one shows that Prop. 2.20
does not work without the conditions on the cohom-functor, while the latter
one tells us that it is not necessary condition and the internal (co)hom-
functor for a monoidal subcategory may differ from the internal (co)hom-
functor for the whole category.

##  3 General representation theory

Here we generalise the approach described in [S21, § 3] for the case of
relatively (co)closed monoidal categories.

###  3.1 Representations in relatively closed monoidal categories

By following the ideas of [S21, § 3.2] we first consider the categories of
(co)representations.

3.1.1. Category of representations. Let
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$
be monoid in a relatively closed monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes,I)$ with a parametrising subcategory
${\mathbf{P}}$, where $X\in{\mathbf{C}}$.

######  Definition 3.1.

A representation of the monoid ${\mathbb{M}}$ on an object $V\in{\mathbf{P}}$
is a morphism $\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$
in the category $\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$. This is a
morphism $\rho\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V)$ in
${\mathbf{C}}$ such that the diagrams

(3.9)

are commutative. Let $\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$ and $\tau\colon{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(W)$ be representations of the same monoid ${\mathbb{M}}$ on the
objects $V\in{\mathbf{P}}$ and $W\in{\mathbf{P}}$ respectively. A morphism
between these representations is a morphism $V\to W$ in ${\mathbf{C}}$ making
the following diagram commute:

(3.14)

Denote by $\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ the
category of representations of the monoid ${\mathbb{M}}$ with their morphisms.
The objects of this category are the pairs $(V,\rho)$, where
$V\in{\mathbf{P}}$ and $\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$ is a representation of ${\mathbb{M}}$ on $V$. A morphism
$(V,\rho)\to(W,\tau)$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ is a morphism
$f\colon V\to W$ in ${\mathbf{P}}$ respecting the representations in the sense
of diagram (3.14).

######  Proposition 3.2.

The forgetful functor

$\displaystyle
G\colon\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\to{\mathbf{P}},$
$\displaystyle(V,\rho)\mapsto V,$ $\displaystyle f\mapsto f,$ (3.15)

reflects isomorphisms: if $(V,\rho)$, $(W,\rho)$ are objects of the category
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ and a morphism
$f\colon(V,\rho)\to(W,\tau)$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ is given by an
isomorphism $f\colon V\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}W$ in
${\mathbf{P}}$, then $f\colon(V,\rho)\to(W,\tau)$ is an isomorphism in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$.

Proof. Let $f^{-1}\colon W\to V$ be the inverse morphism for $f$. Then, by
composing the diagram (3.14) with $\mathop{\mathrm{\bf
hom}}\nolimits(f^{-1},f^{-1})\colon\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\to\mathop{\mathrm{\bf hom}}\nolimits(W,V)$ we see that
$f^{-1}$ gives a morphism $(W,\tau)\to(V,\rho)$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$. ∎

######  Remark 3.3.

Consider the composition functor

$\displaystyle\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})^{\mathrm{op}}\times\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\xrightarrow{G^{\mathrm{op}}\times
G}{\mathbf{P}}^{\mathrm{op}}\times{\mathbf{P}}\xrightarrow{\mathop{\mathrm{\bf
hom}}\nolimits}{\mathbf{C}},$ (3.16)

where $G$ is the forgetful functor (3.15). The commutative diagram (3.14)
means exactly that the formula $\alpha_{(V,\rho)}=\rho\colon
X\to\mathop{\mathrm{\bf hom}}\nolimits(V,V)$ defines a dinatural
transformation $\alpha$ from the constant functor $X$ to (3.16).

Let ${\mathbf{C}}=({\mathbf{C}},\otimes,I)$ be a relatively coclosed monoidal
category with the parametrising subcategory ${\mathbf{P}}$. Consider a
comonoid
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$,
where $X\in{\mathbf{C}}$.

######  Definition 3.4.

A corepresentation of the comonoid ${\mathbb{O}}$ on an object
$V\in{\mathbf{P}}$ is a morphism $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits(V)\to{\mathbb{O}}$ in the category
$\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$. This is a morphism
$\omega\colon\mathop{\mathrm{\bf coend}}\nolimits(V)\to X$ in ${\mathbf{C}}$
such that the diagrams

are commutative. Denote by
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})$ the category
whose objects are the pairs $(V,\omega)$, where $V\in{\mathbf{P}}$ and
$\omega$ is a corepresentation of ${\mathbb{O}}$ on $V$, and morphisms
$(V,\omega)\to(W,\nu)$ are morphisms $f\colon V\to W$ in ${\mathbf{P}}$ such
that

(3.21)

The categorical duality allows us to pass from relatively closed categories to
relatively coclosed categories (see p. 2.21). For any monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes)$ we have the correspondence

$\displaystyle\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})=\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}}^{\mathrm{op}})^{\mathrm{op}},$
(3.22)

which identifies the monoid ${\mathbb{M}}=(X,\mu_{X},\eta_{X})$ in
${\mathbf{C}}$ with the comonoid ${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})$
in the opposite monoidal category
${\mathbf{C}}^{\mathrm{op}}=({\mathbf{C}}^{\mathrm{op}},\otimes)$, where
$\Delta_{X}\colon X\to X\otimes X$ and $\varepsilon_{X}\colon X\to
I_{\mathbf{C}}$ are the morphisms $\mu_{X}\colon X\otimes X\to X$ and
$\eta_{X}\colon I_{\mathbf{C}}\to X$ regarded as morphisms in
${\mathbf{C}}^{\mathrm{op}}$. If ${\mathbf{C}}=({\mathbf{C}},\otimes)$ is
symmetric and closed relative to ${\mathbf{P}}$, then the categories of
representations of the monoid ${\mathbb{M}}$ and corepresentations of the
corresponding comonoid ${\mathbb{O}}$ are opposite to each other. A
representation $\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf
end}}\nolimits(V)$ on $V\in{\mathbf{P}}$ is identified with the
corepresentation $\omega\colon\mathop{\mathrm{\bf
coend}}\nolimits(V)\to{\mathbb{O}}$ by the functor (3.22) as morphisms of
monoids and comonoids respectively, where $\mathop{\mathrm{\bf
end}}\nolimits(V)$ and $\mathop{\mathrm{\bf coend}}\nolimits(V)$ are end-
monoid and coend-monoid for ${\mathbf{C}}$ and ${\mathbf{C}}^{\mathrm{op}}$
respectively identified by (3.22). Moreover, if
$\tau\colon{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(W)$ is a
representation of ${\mathbb{M}}$ on $W\in{\mathbf{P}}$ and
$\nu\colon\mathop{\mathrm{\bf coend}}\nolimits(W)\to{\mathbb{O}}$ is the
corresponding corepresentation, then a morphism $f\colon V\to W$ in
${\mathbf{C}}$ is a morphism $(V,\rho)\to(W,\tau)$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ iff it is a
morphism $(W,\nu)\to(V,\omega)$ in
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})$. Thus we have
the isomorphism of categories

$\displaystyle\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})=\mathop{\mathbf{Corep}}\nolimits_{{\mathbf{P}}^{\mathrm{op}}}({\mathbb{O}})^{\mathrm{op}},$
(3.23)

which identifies $(V,\rho)$ with $(V,\omega)$ and $f\colon(V,\rho)\to(W,\tau)$
with $f\colon(W,\nu)\to(V,\omega)$.

3.1.2. Representations as actions. Recall that for any (symmetric) monoidal
category ${\mathbf{C}}=({\mathbf{C}},\otimes)$ the pairs $(V,a)$ of an object
$V\in{\mathbf{C}}$ with an action $a\colon X\otimes V\to V$ of a monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$
form a category denoted by $\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})$.
For a comonoid
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$
the pairs $(V,\delta)$ of $V\in{\mathbf{C}}$ with a coaction $\delta\colon
V\to X\otimes V$ form the category
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$.

######  Theorem 3.5.

Fix a parametrising subcategory ${\mathbf{P}}\subset{\mathbf{C}}$. If
${\mathbf{C}}$ is relatively closed, then for any $V\in{\mathbf{P}}$ the
bijection

$\displaystyle\theta_{X,V,V}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}X\otimes
V,V\big{)}$ (3.24)

establishes one-to-one correspondence between representations
$\rho\colon{\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(V)$ and actions
$a\colon X\otimes V\to V$ of ${\mathbb{M}}$ on $V$, i.e.
$\rho\in\mathop{\mathrm{Hom}}\nolimits\big{(}X,\mathop{\mathrm{\bf
end}}\nolimits(V)\big{)}$ is a representation iff $a=\theta(\rho)$ is an
action. A morphism $f\colon V\to W$ is a morphism of representations
$(V,\rho)\to(W,\tau)$ iff it is a morphism of actions $(V,a)\to(W,b)$, where
$b=\theta(\tau)$. Thus we can consider
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ as a full
subcategory of $\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})$ via the fully
faithful functor

$\displaystyle\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\hookrightarrow\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}}),$
$\displaystyle(V,\rho)\mapsto\big{(}V,\theta(\rho)\big{)}.$ (3.25)

Analogously, if ${\mathbf{C}}$ is relatively coclosed, then the bijections

$\displaystyle\vartheta_{V,V,X}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}\mathop{\mathrm{\bf
coend}}\nolimits(V),X\big{)}\cong\mathop{\mathrm{Hom}}\nolimits\big{(}V,X\otimes
V\big{)},$ $\displaystyle V\in{\mathbf{P}},$ (3.26)

give the fully faithful functor

$\displaystyle\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})\hookrightarrow\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}}),$
$\displaystyle(V,\omega)\mapsto\big{(}V,\vartheta(\omega)\big{)}.$ (3.27)

If ${\mathbf{C}}$ is closed or coclosed (i.e. ${\mathbf{P}}={\mathbf{C}}$),
then the functor (3.25) or, respectively, (3.27) is an equivalence (more
exactly, isomorphism) of categories.

Proof. Repeats the proof of [S21, Lemmas 3.3, 3.5]. ∎

In the relatively closed case the identity morphism

$\displaystyle X=\mathop{\mathrm{\bf
end}}\nolimits(V)\xrightarrow{\mathop{\mathrm{id}}\nolimits}\mathop{\mathrm{\bf
end}}\nolimits(V)$ (3.28)

is a representation of the monoid $\mathop{\mathrm{\bf
end}}\nolimits(V)=(X,c_{V},u_{V})$ on $V\in{\mathbf{P}}$. The corresponding
action is
$\theta(\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{ev}}\nolimits_{V,V}\colon\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes V\to V$, which was mentioned in p. 2.21. In the
relatively coclosed case we have an analogical corepresentation

$\displaystyle\mathop{\mathrm{\bf
coend}}\nolimits(V)\xrightarrow{\mathop{\mathrm{id}}\nolimits}\mathop{\mathrm{\bf
coend}}\nolimits(V)=X$ (3.29)

of the comonoid $\mathop{\mathrm{\bf coend}}\nolimits(V)=(X,d_{V},v_{V})$ on
the object $V\in{\mathbf{P}}$. It corresponds to the coaction
$\vartheta(\mathop{\mathrm{id}}\nolimits)=\mathop{\mathrm{coev}}\nolimits_{V,V}\colon
V\to\mathop{\mathrm{\bf coend}}\nolimits(V)\otimes V$.

###  3.2 Tensor product of representations

The monoidal product of representations (actions) of a monoid ${\mathbb{M}}$
is introduced by means of an additional structure on ${\mathbb{M}}$, namely by
a structure of comonoid which turn ${\mathbb{M}}$ into a bimonoid. In [S21] we
considered monoidal product in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})$ via a monoidal
product in $\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})$. In the present
work we define and study the former monoidal product without the notion of
(co)action, but then we show that these two ways give the same result. Note
also that here we consider more general case then in [S21], the case of
relatively (co)closed monoidal categories.

In this subsection we always suppose that the monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes,I)$ is symmetric. The corresponding
symmetric structure is a natural isomorphism, which we denote by $\sigma$. It
has components $\sigma=\sigma_{X,Y}\colon X\otimes
Y\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}Y\otimes X$. Let
$X_{1},\ldots,X_{n}$ be objects of ${\mathbf{C}}$. We use the following
notation: for $a=1,\ldots,n-1$ we denote by $\sigma^{(a,a+1)}$ the isomorphism
$X_{1}\otimes\cdots\otimes X_{a}\otimes X_{a+1}\otimes\cdots\otimes
X_{n}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}X_{1}\otimes\cdots\otimes
X_{a+1}\otimes X_{a}\otimes\cdots\otimes X_{n}$, where $\sigma$ acts on the
$a$-th and $(a+1)$-st factors, that is

$\sigma^{(a,a+1)}=\mathop{\mathrm{id}}\nolimits^{\otimes(a-1)}\otimes\sigma\otimes\mathop{\mathrm{id}}\nolimits^{\otimes(n-a-1)}=\mathop{\mathrm{id}}\nolimits_{X_{1}}\otimes\cdots\otimes\mathop{\mathrm{id}}\nolimits_{X_{a-1}}\otimes\sigma_{X_{a},X_{a+1}}\otimes\mathop{\mathrm{id}}\nolimits_{X_{a+2}}\otimes\cdots\otimes\mathop{\mathrm{id}}\nolimits_{X_{n}}.$

3.2.1. Monoidal structure on the category of monoids. Recall that the
symmetric structure of ${\mathbf{C}}$ allows to define a monoidal structure on
the category of monoids $\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$. Let us
describe this structure in details.

The monoidal product of two monoids
$(X,\mu_{X},\eta_{X}),(Y,\mu_{Y},\eta_{Y})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$
is the triple $(X\otimes Y,\mu_{X\otimes Y},\eta_{X\otimes Y})$, where
$\mu_{X\otimes Y}$ and $\eta_{X\otimes Y}$ are defined by the commutative
diagrams

(3.38)

One can check that it is also a monoid in ${\mathbf{C}}$. The monoidal product
of morphisms in $\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$ is obtained as
follows.

######  Proposition 3.6.

Let
$(X,\mu_{X},\eta_{X}),(Y,\mu_{Y},\eta_{Y}),(X^{\prime},\mu_{X^{\prime}},\eta_{X^{\prime}}),(Y^{\prime},\mu_{Y^{\prime}},\eta_{Y^{\prime}})\in\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$.
If $f\colon X\to Y$ and $g\colon Y\to Y^{\prime}$ are morphisms of monoids
$(X,\mu_{X},\eta_{X})\to(X^{\prime},\mu_{X^{\prime}},\eta_{X^{\prime}})$ and
$(Y,\mu_{Y},\eta_{Y})\to(Y^{\prime},\mu_{Y^{\prime}},\eta_{Y^{\prime}})$, then
$f\otimes g\colon X\otimes Y\to X^{\prime}\otimes Y^{\prime}$ is a morphism

$(X\otimes Y,\mu_{X\otimes Y},\eta_{X\otimes Y})\to(X^{\prime}\otimes
Y^{\prime},\mu_{X^{\prime}\otimes Y^{\prime}},\eta_{X^{\prime}\otimes
Y^{\prime}}).$

Proof. It follows from the commutative diagrams

and the definitions (3.38). ∎

Thus we have a bifunctor
$-\otimes-\colon\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\times\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})\to\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$.
It is easy to check that
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})=\big{(}\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}}),\otimes\big{)}$
is a symmetric monoidal category with the unit object
$I_{\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})}=(I,\mathop{\mathrm{id}}\nolimits_{I},\mathop{\mathrm{id}}\nolimits_{I})$.
The category of comonoids $\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})$ is
equipped with a structure of symmetric monoidal category in the dual way.

3.2.2. Natural transformation $\pi$. Until the end of this subsection we
suppose that the subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ is monoidal
(this condition is reduced to $I\in{\mathbf{P}}$, $V\otimes
V^{\prime}\in{\mathbf{P}}\;\forall\,V,V^{\prime}\in{\mathbf{P}}$, since
${\mathbf{P}}$ is a full subcategory of ${\mathbf{C}}$). Then for any
$V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$ we have the hom-object
$\mathop{\mathrm{\bf hom}}\nolimits(V\otimes V^{\prime},W\otimes W^{\prime})$.
In particular, we have the monoids $\mathop{\mathrm{\bf
end}}\nolimits(V\otimes V^{\prime})$ and $\mathop{\mathrm{\bf
end}}\nolimits(I)$. By Prop. 2.25 the latter monoid is the unit object:
$\mathop{\mathrm{\bf
end}}\nolimits(I)=(I,\mathop{\mathrm{id}}\nolimits_{I},\mathop{\mathrm{id}}\nolimits_{I})$.
We are going to construct some useful morphisms of monoids
$\mathop{\mathrm{\bf end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime})\to\mathop{\mathrm{\bf end}}\nolimits(V\otimes
V^{\prime})$.

Let us use the notation $[V,W]:=\mathop{\mathrm{\bf hom}}\nolimits(V,W)$ for
briefness. Consider the composition

$\displaystyle[V,W]\otimes[V^{\prime},W^{\prime}]\otimes V\otimes
V^{\prime}\xrightarrow{\sigma^{(23)}}[V,W]\otimes
V\otimes[V^{\prime},W^{\prime}]\otimes
V^{\prime}\xrightarrow{\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits}W\otimes
W^{\prime},$ (3.39)

where $V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$. By applying the bijection

$\theta^{-1}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}[V,W]\otimes[V^{\prime},W^{\prime}]\otimes
V\otimes V^{\prime},W\otimes
W^{\prime}\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\\\
\mathop{\mathrm{Hom}}\nolimits\big{(}[V,W]\otimes[V^{\prime},W^{\prime}],[V\otimes
V^{\prime},W\otimes W^{\prime}]\big{)}$

to (3.39) we get the morphism
$\pi=\pi^{V,V^{\prime}}_{W,W^{\prime}}=\theta^{-1}\big{(}(\mathop{\mathrm{ev}}\nolimits_{V,W}\otimes\mathop{\mathrm{ev}}\nolimits_{V^{\prime},W^{\prime}})\cdot\sigma^{(23)}\big{)}$
of the form

$\displaystyle\pi\colon[V,W]\otimes[V^{\prime},W^{\prime}]\to[V\otimes
V^{\prime},W\otimes W^{\prime}].$ (3.40)

This is a unique morphism making the diagram

(3.45)

commute.

######  Proposition 3.7.

The morphisms $\pi^{V,V^{\prime}}_{W,W^{\prime}}$ are natural in all four
arguments $V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$.

Proof. We need to prove the commutativity of the diagram

(3.50)

for arbitrary morphisms $f\colon U\to V$, $f\colon U^{\prime}\to V^{\prime}$,
$g\colon W\to Z$ and $g^{\prime}\colon W^{\prime}\to Z^{\prime}$ in
${\mathbf{P}}$. Let us apply $\theta$ to the compositions in this diagram.
This amounts to monoidal multiplication of each object and arrow in the
diagram by $U\otimes U^{\prime}$ from the right and composition with
$\mathop{\mathrm{ev}}\nolimits\colon[U\otimes U^{\prime},Z\otimes
Z^{\prime}]\otimes U\otimes U^{\prime}\to Z\otimes Z^{\prime}$. By using the
commutativity of (3.45) and naturality of $\sigma$ we get the equivalent
diagram

By virtue of Prop. 2.24 we obtain

$\displaystyle\mathop{\mathrm{ev}}\nolimits\cdot\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f\otimes f^{\prime},g\otimes
g^{\prime})\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}}\big{)}=\theta\big{(}\mathop{\mathrm{\bf hom}}\nolimits(f\otimes
f^{\prime},g\otimes g^{\prime})\big{)}=(g\otimes
g^{\prime})\cdot\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\otimes
f\otimes f^{\prime}),$
$\displaystyle(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f,g)\otimes\mathop{\mathrm{id}}\nolimits_{U}\otimes\mathop{\mathrm{\bf
hom}}\nolimits(f^{\prime},g^{\prime})\otimes\mathop{\mathrm{id}}\nolimits_{U^{\prime}}\big{)}=\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f,g)\big{)}\otimes\theta\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(f^{\prime},g^{\prime})\big{)}=$
$\displaystyle\big{(}g\cdot\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\otimes
f)\big{)}\otimes\big{(}g^{\prime}\cdot\mathop{\mathrm{ev}}\nolimits\cdot(\mathop{\mathrm{id}}\nolimits\otimes
f^{\prime})\big{)},$

so the diagram (3.50) is equivalent to

(3.57)

By drawing the arrow $\sigma^{(23)}$ from the top right corner to the down
left one we see that the commutativity of the diagram (3.57) follows from the
naturality of $\sigma$ and commutativity of (3.45). ∎

3.2.3. Properties of $\pi$. The natural transformation $\pi$ is agreed with
the internal composition (2.41) in the following sense.

######  Proposition 3.8.

For any objects $U,V,W,U^{\prime},V^{\prime},W^{\prime}\in{\mathbf{P}}$ we get
a commutative diagram

where $c_{U,V,W}$ are the internal compositions (2.41).

Proof. We need to show the equality of two morphisms $X\to[U\otimes
U^{\prime},W\otimes W^{\prime}]$, where
$X=[V,W]\otimes[V^{\prime},W^{\prime}]\otimes[U,V]\otimes[U^{\prime},V^{\prime}]$.
One can show that their images under $\theta_{X,U\otimes U^{\prime},W\otimes
W^{\prime}}$ coincide with each other by using the naturality of $\sigma$ and
the diagrams (3.45), (2.46) in an appropriate way. However, the proof in terms
of diagrams would take too much space. We present it more compactly. By using
the formula (2.39) and the definition of $c_{U\otimes U^{\prime},V\otimes
V^{\prime},W\otimes W^{\prime}}$ we obtain

$\theta\big{(}c_{U\otimes U^{\prime},V\otimes V^{\prime},W\otimes
W^{\prime}}\cdot(\pi\otimes\pi)\big{)}=\theta(c_{U\otimes U^{\prime},V\otimes
V^{\prime},W\otimes
W^{\prime}})\cdot(\pi\otimes\pi\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}})=\\\ \mathop{\mathrm{ev}}\nolimits_{V\otimes V^{\prime},W\otimes
W^{\prime}}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U\otimes
U^{\prime},V\otimes
V^{\prime}})\cdot(\pi\otimes\pi\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}}).$ (3.58)

Note that
$(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U\otimes
U^{\prime},V\otimes
V^{\prime}})\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits)=\pi\otimes\mathop{\mathrm{ev}}\nolimits_{U\otimes
U^{\prime},V\otimes
V^{\prime}}=(\pi\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U\otimes
U^{\prime},V\otimes V^{\prime}})$, so (3.58) equals

$\mathop{\mathrm{ev}}\nolimits_{V\otimes V^{\prime},W\otimes
W^{\prime}}\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U\otimes
U^{\prime},V\otimes
V^{\prime}})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\pi\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}})=\\\
(\mathop{\mathrm{ev}}\nolimits_{V,W}\otimes\mathop{\mathrm{ev}}\nolimits_{V^{\prime},W^{\prime}})\cdot\sigma^{(23)}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U,V}\otimes\mathop{\mathrm{ev}}\nolimits_{U^{\prime},V^{\prime}})\cdot\sigma^{(45)},$
(3.59)

where we used the commutative diagram (3.45). Then, by taking into account the
equality
$\sigma^{(23)}\cdot(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U,V}\otimes\mathop{\mathrm{ev}}\nolimits_{U^{\prime},V^{\prime}})=(\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U,V}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits_{U^{\prime},V^{\prime}})\cdot\sigma^{(34)}\cdot\sigma^{(23)}$
and again the definition of internal composition we can rewrite (3.59) in the
form

$(\mathop{\mathrm{ev}}\nolimits_{U,W}\otimes\mathop{\mathrm{ev}}\nolimits_{U^{\prime},W^{\prime}})\cdot(c_{U,V,W}\otimes\mathop{\mathrm{id}}\nolimits_{U}\otimes
c_{U^{\prime},V^{\prime},W^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{U^{\prime}})\cdot\sigma^{(34)}\cdot\sigma^{(23)}\cdot\sigma^{(45)}=\\\
(\mathop{\mathrm{ev}}\nolimits_{U,W}\otimes\mathop{\mathrm{ev}}\nolimits_{U^{\prime},W^{\prime}})\cdot\sigma^{(23)}\cdot(c_{U,V,W}\otimes
c_{U^{\prime},V^{\prime},W^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{U}\otimes\mathop{\mathrm{id}}\nolimits_{U^{\prime}})\cdot\sigma^{(23)}=\\\
\theta(\pi)\cdot(c_{U,V,W}\otimes
c_{U^{\prime},V^{\prime},W^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}})\cdot(\sigma^{(23)}\otimes\mathop{\mathrm{id}}\nolimits_{U\otimes
U^{\prime}})=\theta\big{(}\pi\cdot(c_{U,V,W}\otimes
c_{U^{\prime},V^{\prime},W^{\prime}})\cdot\sigma^{(23)}\big{)}.$

Since $\theta$ is a bijection, the proposition is proved. ∎

Let us establish ‘associativity’ of the natural transformation $\pi$.

######  Proposition 3.9.

For any
$V,W,V^{\prime},W^{\prime},V^{\prime\prime},W^{\prime\prime}\in{\mathbf{P}}$
we have the commutative diagram

(3.64)

Proof. By using the diagram (3.45) one can show that the morphism
$\theta\big{(}\pi\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits)\big{)}$
coincides with the composition
$[V,W]\otimes[V^{\prime},W^{\prime}]\otimes[V^{\prime\prime},W^{\prime\prime}]\otimes
V\otimes V^{\prime}\otimes
V^{\prime\prime}\xrightarrow{\sigma^{(23)}\sigma^{(45)}\sigma^{(34)}}[V,W]\otimes
V\otimes[V^{\prime},W^{\prime}]\otimes
V^{\prime}\otimes[V^{\prime\prime},W^{\prime\prime}]\otimes
V^{\prime\prime}\xrightarrow{\mathop{\mathrm{ev}}\nolimits_{V,W}\otimes\mathop{\mathrm{ev}}\nolimits_{V^{\prime},W^{\prime}}\otimes\mathop{\mathrm{ev}}\nolimits_{V^{\prime\prime},W^{\prime\prime}}}W\otimes
W^{\prime}\otimes W^{\prime\prime}$. Analogously, one derives that
$\theta\big{(}\pi\cdot(\mathop{\mathrm{id}}\nolimits\otimes\pi)\big{)}$
coincides with the same composition. ∎

######  Proposition 3.10.

The natural transformation $\pi$ is in agreement with the symmetric structure
$\sigma$ in the sense that the diagram

(3.69)

commutes, where

$\displaystyle\sigma=\sigma_{[V,W],[V^{\prime},W^{\prime}]}\colon[V,W]\otimes[V^{\prime},W^{\prime}]\to[V^{\prime},W^{\prime}]\otimes[V,W],$
(3.70) $\displaystyle\sigma_{V^{\prime},V}\colon V^{\prime}\otimes V\to
V\otimes V^{\prime},\quad\quad\sigma_{W,W^{\prime}}\colon W\otimes
W^{\prime}\to W^{\prime}\otimes W$ (3.71)

are the corresponding components of the natural transformation $\sigma$.

Proof. Consider the morphism
$\sigma^{-1}=\sigma_{[V^{\prime},W^{\prime}],[V,W]}$ inverse to (3.70).
Together with the morphisms (3.71) and naturality of $\theta^{-1}$ it gives
the commutative diagram

(3.76)

Recall that
$\pi=\theta\big{(}(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}\big{)}$.
The commutativity of the diagram

(3.81)

implies that the left vertical arrow of the diagram (3.76) maps the morphism
$(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}$
to
$(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}$.
Hence, by applying (3.76) to this morphism we obtain the needed equality
$\pi=\mathop{\mathrm{\bf
hom}}\nolimits(\sigma_{V^{\prime},V},\sigma_{W,W^{\prime}})\cdot\pi\cdot\sigma^{-1}$.
∎

3.2.4. The morphisms $\pi$ for internal ends. If $W=V$ and
$W^{\prime}=V^{\prime}$, then the morphism (3.40) takes the form

$\displaystyle\pi\colon\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime})\to\mathop{\mathrm{\bf end}}\nolimits(V\otimes
V^{\prime}).$ (3.82)

The domain object of this morphism is equipped with a structure of monoid as a
monoidal product of two monoids; namely, the multiplication morphism for this
monoid is $(c_{V}\otimes c_{V^{\prime}})\cdot\sigma^{(23)}$, while the unit is
$u_{V}\otimes u_{V^{\prime}}$. The codomain of (3.82) is a monoid with the
structure $(c_{V\otimes V^{\prime}},u_{V\otimes V^{\prime}})$. Prop. 3.8 for
$U=W=V$ and $U^{\prime}=W^{\prime}=V^{\prime}$ means exactly that (3.82)
preserves the multiplications. Let us prove that it also preserves the units.

######  Lemma 3.11.

The following diagram is commutative:

(3.87)

Proof. Due to the formula (2.38) and commutativity of the diagram (3.45) we
obtain

$\theta\big{(}\pi\cdot(u_{V}\otimes
u_{V^{\prime}})\big{)}=\mathop{\mathrm{ev}}\nolimits_{V\otimes
V^{\prime},V\otimes
V^{\prime}}\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}})\cdot(u_{V}\otimes
u_{V^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes V^{\prime}})=\\\
(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}\cdot(u_{V}\otimes
u_{V^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{V}\otimes\mathop{\mathrm{id}}\nolimits_{V^{\prime}})=(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot(u_{V}\otimes\mathop{\mathrm{id}}\nolimits_{V}\otimes
u_{V^{\prime}}\otimes\mathop{\mathrm{id}}\nolimits_{V^{\prime}})\cdot\sigma^{(23)}.$
(3.88)

From the diagram (2.65) we see that the right hand side of the formula (3.88)
is the identification $I\otimes I\otimes V\otimes V^{\prime}=V\otimes
V^{\prime}$, so it equals $\theta(u_{V\otimes V^{\prime}})$. ∎

######  Proposition 3.12.

The morphisms (3.82) are morphisms of monoids.

Proof. This is a direct consequence of Prop. 3.8 and Lemma 3.11. ∎

######  Proposition 3.13.

Both morphisms

$\displaystyle\pi^{I,V}_{I,W}\colon[I,I]\otimes[V,W]\to[I\otimes V,I\otimes
W],$ $\displaystyle\pi^{V,I}_{W,I}\colon[V,W]\otimes[I,I]\to[V\otimes
I,W\otimes I]$

are equal to $\mathop{\mathrm{id}}\nolimits_{[V,W]}$. In particular, the
morphism (3.82) for $V=I$ or $V^{\prime}=I$ takes the form of the
identification $\mathop{\mathrm{\bf
end}}\nolimits(I)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime})=I\otimes\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime})=\mathop{\mathrm{\bf end}}\nolimits(V^{\prime})$ or,
respectively, it coincides with $\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(I)=\mathop{\mathrm{\bf end}}\nolimits(V)\otimes
I=\mathop{\mathrm{\bf end}}\nolimits(V)$.

Proof. By substituting $V^{\prime}=W^{\prime}=I$ (or $V=W=I$) to the diagram
(3.45) and using the identifications $I\otimes V=V=I\otimes V$, $[I,I]=I$ we
see that $\pi=\mathop{\mathrm{id}}\nolimits_{[V,W]}$ makes it commute. Then
the statement of the proposition follows from the uniqueness of $\pi$. ∎

######  Lemma 3.14.

The morphisms (3.82) make the following diagram commute:

(3.93)

Proof. This follows from Prop. 3.10 and the equalities $\mathop{\mathrm{\bf
hom}}\nolimits(\sigma_{V,V^{\prime}},\mathop{\mathrm{id}}\nolimits)\cdot\mathop{\mathrm{\bf
hom}}\nolimits(\sigma_{V^{\prime},V},\sigma_{V,V^{\prime}})=\mathop{\mathrm{\bf
hom}}\nolimits(\sigma_{V,V^{\prime}}\cdot\sigma_{V^{\prime},V},\sigma_{V,V^{\prime}})=\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,\sigma_{V,V^{\prime}})$. ∎

3.2.5. Natural transformation $\varkappa$. In the relatively coclosed case we
can consider cohom-objects $\mathop{\mathrm{\bf cohom}}\nolimits(V\otimes
V^{\prime},W\otimes W^{\prime})$ for any objects
$V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$. In particular, the have comonoids
$\mathop{\mathrm{\bf coend}}\nolimits(V\otimes V^{\prime})$ and
$\mathop{\mathrm{\bf
coend}}\nolimits(I)=I_{\mathop{\mathbf{Comon}}\nolimits({\mathbf{C}})}=(I,\mathop{\mathrm{id}}\nolimits_{I},\mathop{\mathrm{id}}\nolimits_{I})$.

To define morphisms dual to (3.40) we apply the bijection

$\vartheta^{-1}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}W\otimes
W^{\prime},\mathop{\mathrm{\bf cohom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime})\otimes V\otimes
V^{\prime}\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\\\
\mathop{\mathrm{Hom}}\nolimits\big{(}\mathop{\mathrm{\bf
cohom}}\nolimits(V\otimes V^{\prime},W\otimes W^{\prime}),\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime})\big{)}$

to the composition

$W\otimes
W^{\prime}\xrightarrow{\mathop{\mathrm{coev}}\nolimits\otimes\mathop{\mathrm{coev}}\nolimits}\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes V\otimes\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime})\otimes
V^{\prime}\xrightarrow{\sigma^{(23)}}\\\ \mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime})\otimes V\otimes V^{\prime}.$

This gives the morphism
$\varkappa=\varkappa^{V,V^{\prime}}_{W,W^{\prime}}=\theta^{-1}\big{(}\sigma^{(23)}\cdot(\mathop{\mathrm{coev}}\nolimits\otimes\mathop{\mathrm{coev}}\nolimits)\big{)}\colon\mathop{\mathrm{\bf
cohom}}\nolimits(V\otimes V^{\prime},W\otimes
W^{\prime})\to\mathop{\mathrm{\bf
cohom}}\nolimits(V,W)\otimes\mathop{\mathrm{\bf
cohom}}\nolimits(V^{\prime},W^{\prime}).$ By applying duality principle to
Prop. 3.7 we conclude that these morphisms are also natural in
$V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$.

The natural transformation $\varkappa$ is coassociative in the sense similar
to Prop. 3.9, but with the opposite arrows:
$(\varkappa\otimes\mathop{\mathrm{id}}\nolimits)\cdot\varkappa=(\mathop{\mathrm{id}}\nolimits\otimes\varkappa)\cdot\varkappa$.

In the case $W=V$ and $W^{\prime}=V^{\prime}$ it reads

$\displaystyle\varkappa\colon\mathop{\mathrm{\bf coend}}\nolimits(V\otimes
V^{\prime})\to\mathop{\mathrm{\bf
coend}}\nolimits(V)\otimes\mathop{\mathrm{\bf coend}}\nolimits(V^{\prime}).$
(3.94)

######  Proposition 3.15.

The morphisms (3.94) are morphisms of comonoids.

Proof. It follows from Prop. 3.12 by the duality principle. ∎

3.2.6. Tensor product of representations of bimonoids. A bimonoid in the
symmetric monoidal category ${\mathbf{C}}$ is
${\mathbb{B}}=(X,\mu_{X},\eta_{X},\Delta_{X},\varepsilon_{X})\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}})$,
where $\mu_{X}$, $\eta_{X}$, $\Delta_{X}$ and $\varepsilon_{X}$ are
multiplication, unit, comultiplication and counit respectively on the object
$X\in{\mathbf{C}}$. By considering ${\mathbb{B}}$ as a monoid
${\mathbb{M}}=(X,\mu_{X},\eta_{X})$ we obtain the category
$\mathop{\mathbf{Lact}}\nolimits({\mathbb{B}})=\mathop{\mathbf{Lact}}\nolimits({\mathbb{M}})$.
If ${\mathbf{C}}$ is closed relative to ${\mathbf{P}}$, then we have the
subcategory
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})=\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{M}})\subset\mathop{\mathbf{Lact}}\nolimits({\mathbb{B}})$.
Here we describe how the comonoid structure $(\Delta_{X},\varepsilon_{X})$
gives a monoidal structure on the category
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$.

Let $\rho\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V)$ and
$\rho^{\prime}\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V^{\prime})$ be
two representations of the bimonoid ${\mathbb{B}}$ (i.e. of the monoid
${\mathbb{M}}$) on the objects $V,V^{\prime}\in{\mathbf{P}}$. Define a
morphism $\rho^{\prime\prime}\colon X\to\mathop{\mathrm{\bf
end}}\nolimits(V\otimes V^{\prime})$ as the composition

$\displaystyle X\xrightarrow{\Delta_{X}}X\otimes
X\xrightarrow{\rho\otimes\rho^{\prime}}\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime})\xrightarrow{\pi}\mathop{\mathrm{\bf
end}}\nolimits(V\otimes V^{\prime}).$ (3.95)

######  Lemma 3.16.

The morphism (3.95) is a representation of ${\mathbb{B}}$ on $V\otimes
V^{\prime}$.

Proof. Since ${\mathbb{B}}$ is a bimonoid, $\Delta_{X}$ is a morphism of
monoids ${\mathbb{M}}\to{\mathbb{M}}\otimes{\mathbb{M}}$. From Prop. 3.6 we
obtain the monoid morphism
${\mathbb{M}}\otimes{\mathbb{M}}\xrightarrow{\rho\otimes\rho^{\prime}}\mathop{\mathrm{\bf
end}}\nolimits(V)\otimes\mathop{\mathrm{\bf end}}\nolimits(V^{\prime})$. Due
to Prop. 3.12 the formula (3.95) gives a morphism
${\mathbb{M}}\to\mathop{\mathrm{\bf end}}\nolimits(V\otimes V^{\prime})$ in
$\mathop{\mathbf{Mon}}\nolimits({\mathbf{C}})$. ∎

We call the representation (3.95) the tensor product of the representations
$\rho$, $\rho^{\prime}$.

######  Lemma 3.17.

Let $\rho$, $\rho^{\prime}$, $\tau$, $\tau^{\prime}$ be representations of
${\mathbb{B}}$ on $V,V^{\prime},W,W^{\prime}\in{\mathbf{P}}$. Denote the
tensor products of $\rho$, $\rho^{\prime}$ and of $\tau$, $\tau^{\prime}$ by
$\rho^{\prime\prime}$ and by $\tau^{\prime\prime}$ respectively. If morphisms
$f\colon V\to W$ and $f^{\prime}\colon V^{\prime}\to W^{\prime}$ are morphisms
of representations, i.e. morphisms $f\colon(V,\rho)\to(W,\tau)$ and
$f^{\prime}\colon(V^{\prime},\rho^{\prime})\to(W^{\prime},\tau^{\prime})$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$, then $f\otimes
f^{\prime}\colon V\otimes V^{\prime}\to W\otimes W^{\prime}$ is a morphism
$f\otimes f^{\prime}\colon(V\otimes
V^{\prime},\rho^{\prime\prime})\to(W\otimes W^{\prime},\tau^{\prime\prime})$
in $\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$.

Proof. By definition 3.1 the fact that $f$ and $f^{\prime}$ are morphisms of
representations means the equalities $\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,f)\cdot\rho=\mathop{\mathrm{\bf
hom}}\nolimits(f,\mathop{\mathrm{id}}\nolimits)\cdot\tau$ and
$\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,f^{\prime})\cdot\rho^{\prime}=\mathop{\mathrm{\bf
hom}}\nolimits(f^{\prime},\mathop{\mathrm{id}}\nolimits)\cdot\tau^{\prime}$.
By taking into account these equalities and the naturality of $\pi$ (Prop.
3.7) we obtain the commutative diagram

Composition with $\Delta_{X}\colon X\to X\otimes X$ gives $\mathop{\mathrm{\bf
hom}}\nolimits(\mathop{\mathrm{id}}\nolimits,f\otimes
f^{\prime})\cdot\rho^{\prime\prime}=\mathop{\mathrm{\bf
hom}}\nolimits(f\otimes
f^{\prime},\mathop{\mathrm{id}}\nolimits)\cdot\tau^{\prime\prime}$. ∎

######  Theorem 3.18.

For any bimonoid ${\mathbb{B}}$ in a symmetric monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes)$ which is closed relative to a monoidal
subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ the category
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is equipped with
the monoidal structure in the following way. The monoidal product of objects
$(V,\rho)$ and $(V^{\prime},\rho^{\prime})$ is the object
$(V^{\prime\prime},\rho^{\prime\prime})$, where $V^{\prime\prime}=V\otimes
V^{\prime}$ and $\rho^{\prime\prime}\colon{\mathbb{B}}\to\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime\prime})$ is defined by (3.95). The monoidal product
of morphisms $f$ and $g$ in
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ coincides with
their monoidal product in ${\mathbf{P}}$. The unit object of
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is
$(I_{\mathbf{C}},\varepsilon_{X})$. If ${\mathbb{B}}$ is cocommutative (i.e.
$\sigma\cdot\Delta_{X}=\Delta_{X}$), then the monoidal category
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is symmetric and
$\sigma_{(V,\rho),(V^{\prime},\rho^{\prime})}=\sigma_{V,V^{\prime}}$.

Proof. The monoidal operation on objects and on morphisms of
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is defined
correctly due to Lemmas 3.16 and 3.17 respectively. Let us check its
associativity. Consider representations $\rho,\rho^{\prime},\widetilde{\rho}$
of ${\mathbb{B}}$ on $V,V^{\prime},\widetilde{V}\in{\mathbf{P}}$ respectively
and let $\rho^{\prime\prime}$ be the tensor product of $\rho$ and
$\rho^{\prime}$. The sequential monoidal products of objects $(V,\rho)$,
$(V^{\prime},\rho^{\prime})$ and $(\widetilde{V},\widetilde{\rho})$ is given
by the tensor product of the representations $\rho^{\prime\prime}$ and
$\widetilde{\rho}$, this is a representation of ${\mathbb{B}}$ on $V\otimes
V^{\prime}\otimes\widetilde{V}$ of the form

$\pi\cdot(\rho^{\prime\prime}\otimes\widetilde{\rho})\cdot\Delta_{X}=\pi\cdot(\mathop{\mathrm{id}}\nolimits\otimes\widetilde{\rho})\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits)\cdot(\rho\otimes\rho^{\prime}\otimes\mathop{\mathrm{id}}\nolimits)\cdot(\Delta_{X}\otimes\mathop{\mathrm{id}}\nolimits)\cdot\Delta_{X}=\\\
\pi\cdot(\pi\otimes\mathop{\mathrm{id}}\nolimits)\cdot(\rho\otimes\rho^{\prime}\otimes\widetilde{\rho})\cdot(\Delta_{X}\otimes\mathop{\mathrm{id}}\nolimits)\cdot\Delta_{X}.$
(3.96)

By taking the tensor products in another order we obtain

$\pi\cdot(\rho\otimes\widehat{\rho})\cdot\Delta_{X}=\pi\cdot(\rho\otimes\mathop{\mathrm{id}}\nolimits)\cdot(\mathop{\mathrm{id}}\nolimits\otimes\pi)\cdot(\mathop{\mathrm{id}}\nolimits\otimes\rho^{\prime}\otimes\widetilde{\rho})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\Delta_{X})\cdot\Delta_{X}=\\\
\pi\cdot(\mathop{\mathrm{id}}\nolimits\otimes\pi)\cdot(\rho\otimes\rho^{\prime}\otimes\widetilde{\rho})\cdot(\mathop{\mathrm{id}}\nolimits\otimes\Delta_{X})\cdot\Delta_{X},$
(3.97)

where $\widehat{\rho}$ is the tensor product of $\rho^{\prime}$ and
$\widetilde{\rho}$. The representation (3.96) coincides with (3.97) by virtue
of coassociativity of $\Delta_{X}$ and Prop. 3.9. The associativity of the
monoidal operation on morphisms follows from the associativity of the monoidal
product in ${\mathbf{P}}$.

By definition of bimonoid the counit $\varepsilon_{X}$ is in agreement with
the monoid structure $(\mu_{X},\eta_{X})$, i.e. the diagrams

commute. This means exactly that $\varepsilon_{X}\colon X\to
I=\mathop{\mathrm{\bf end}}\nolimits(I)$ is a representation of ${\mathbb{B}}$
on the object $I$. Let us check that
$(I,\varepsilon_{X})\in\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$
is a unit object. The tensor product of a representation $\rho\colon
X\to\mathop{\mathrm{\bf end}}\nolimits(V)$ with the representation
$\varepsilon_{X}\colon X\to I=\mathop{\mathrm{\bf end}}\nolimits(I)$ equals
$\pi^{I,V}_{I,V}\cdot(\varepsilon_{X}\otimes\rho)\cdot\Delta_{X}=\pi^{I,V}_{I,V}\cdot(\mathop{\mathrm{id}}\nolimits_{I}\otimes\rho)\cdot(\varepsilon_{X}\otimes\mathop{\mathrm{id}}\nolimits_{X})\cdot\Delta_{X}$.
Due to
$(\varepsilon_{X}\otimes\mathop{\mathrm{id}}\nolimits_{X})\cdot\Delta_{X}=\mathop{\mathrm{id}}\nolimits_{X}$
and Prop. 3.13 it coincides with $\rho$, so that
$(I,\varepsilon_{X})\otimes(V,\rho)=(V,\rho)$. The identification
$(V,\rho)\otimes(I,\varepsilon_{X})=(V,\rho)$ is checked in the same way.

If ${\mathbb{B}}$ is cocommutative, then due to the naturality of $\sigma$ and
Lemma 3.14 the diagram

commutes, so $\sigma_{V,V^{\prime}}\colon V\otimes
V^{\prime}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}V^{\prime}\otimes
V$ is a morphism
$(V,\rho)\otimes(V^{\prime},\rho^{\prime})\to(V^{\prime},\rho^{\prime})\otimes(V,\rho)$
in $\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$. This is an
isomorphism in $\mathop{\mathbf{Rep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$,
since its inverse $\sigma_{V^{\prime},V}$ is a morphism
$(V^{\prime},\rho^{\prime})\otimes(V,\rho)\to(V,\rho)\otimes(V^{\prime},\rho^{\prime})$.
∎

Let us draw the dual picture. The concept of bimonoid is self-dual. By
considering ${\mathbb{B}}$ as a comonoid
${\mathbb{O}}=(X,\Delta_{X},\varepsilon_{X})$ we obtain the category
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{B}})=\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{O}})$.
If ${\mathbf{C}}$ is coclosed relative to ${\mathbf{P}}$, then we define
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})=\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{O}})\subset\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{B}})$.
This subcategory is a monoidal category whose monoidal structure is given by
the morphisms $\mu_{X},\eta_{X}$. To describe this structure we dualise Lemma
3.16 and Theorem 3.18.

######  Lemma 3.19.

Let $\omega\colon\mathop{\mathrm{\bf coend}}\nolimits(V)\to X$ and
$\omega^{\prime}\colon\mathop{\mathrm{\bf coend}}\nolimits(V^{\prime})\to X$
be corepresentations of the bimonoid ${\mathbb{B}}$ (as a comonoid
${\mathbb{O}}$) on the objects $V\in{\mathbf{P}}$ and
$V^{\prime}\in{\mathbf{P}}$ respectively. Then the morphism
$\omega^{\prime\prime}\colon\mathop{\mathrm{\bf coend}}\nolimits(V\otimes
V^{\prime})\to X$ defined as the composition

$\displaystyle\mathop{\mathrm{\bf coend}}\nolimits(V\otimes
V^{\prime})\xrightarrow{\varkappa}\mathop{\mathrm{\bf
coend}}\nolimits(V)\otimes\mathop{\mathrm{\bf
coend}}\nolimits(V^{\prime})\xrightarrow{\omega\otimes\omega^{\prime}}X\otimes
X\xrightarrow{\mu_{X}}X$ (3.98)

is a corepresentation of ${\mathbb{B}}$ on $V\otimes V^{\prime}$.

The corepresentation (3.98) is called the tensor product of the
corepresentations $\omega,\omega^{\prime}$.

######  Theorem 3.20.

For any bimonoid ${\mathbb{B}}$ in a symmetric monoidal category
${\mathbf{C}}=({\mathbf{C}},\otimes)$ which is coclosed relative to a monoidal
subcategory ${\mathbf{P}}\subset{\mathbf{C}}$ the category
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is equipped with
the monoidal structure. The monoidal product on objects has the form
$(V,\omega)\otimes(V^{\prime},\omega^{\prime})=(V^{\prime\prime},\omega^{\prime\prime})$,
where $V^{\prime\prime}=V\otimes V^{\prime}$ and
$\omega^{\prime\prime}\colon\mathop{\mathrm{\bf
coend}}\nolimits(V^{\prime\prime})\to{\mathbb{B}}$ is the corepresentation
(3.98). On morphisms it is given by the monoidal product $\otimes$ of
${\mathbf{C}}$. The unit object of
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is
$(I_{\mathbf{C}},\eta_{X})$. If ${\mathbb{B}}$ is commutative (i.e.
$\mu_{X}\cdot\sigma=\mu_{X}$), then the monoidal category
$\mathop{\mathbf{Corep}}\nolimits_{\mathbf{P}}({\mathbb{B}})$ is symmetric and
$\sigma_{(V,\omega),(V^{\prime},\omega^{\prime})}=\sigma_{V,V^{\prime}}$.

3.2.7. Tensor product of actions. Since Theorem 3.5 identifies the
representations with actions, Lemma 3.16 allows to define tensor product of
actions of the bimonoid ${\mathbb{B}}$ on objects of ${\mathbf{P}}$ in the
relatively closed case. This operation can be written explicitly via the
coproduct $\Delta_{X}$. In the relatively coclosed case we have tensor product
of coactions of the bimonoid ${\mathbb{B}}$ on objects of ${\mathbf{P}}$,
which can be written via $\mu_{X}$.

######  Proposition 3.21.

Let $\rho\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V)$ and
$\rho^{\prime}\colon X\to\mathop{\mathrm{\bf end}}\nolimits(V^{\prime})$ be
representations of ${\mathbb{B}}$ on $V,V^{\prime}\in{\mathbf{P}}$ and
$\rho^{\prime\prime}\colon X\to\mathop{\mathrm{\bf
end}}\nolimits(V^{\prime\prime})$ be their tensor product (3.95), where
$V^{\prime\prime}=V\otimes V^{\prime}$. Consider the corresponding actions
$a=\theta(\rho)\colon X\otimes V\to V$,
$a^{\prime}=\theta(\rho^{\prime})\colon X\otimes V^{\prime}\to V^{\prime}$ and
$a^{\prime\prime}=\theta(\rho^{\prime\prime})\colon X\otimes
V^{\prime\prime}\to V^{\prime\prime}$. Then $a^{\prime\prime}$ coincides with
the composition

$\displaystyle X\otimes V\otimes
V^{\prime}\xrightarrow{\Delta_{X}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits}X\otimes
X\otimes V\otimes V^{\prime}\xrightarrow{\sigma^{(23)}}X\otimes V\otimes
X\otimes V^{\prime}\xrightarrow{a\otimes a^{\prime}}V\otimes V^{\prime}.$
(3.99)

Dually, let $\omega\colon\mathop{\mathrm{\bf coend}}\nolimits(V)\to X$ and
$\omega^{\prime}\colon\mathop{\mathrm{\bf coend}}\nolimits(V^{\prime})\to X$
be corepresentations of the bimonoid ${\mathbb{B}}$ on the objects
$V,V^{\prime}$. Let $\delta=\vartheta(\omega)\colon V\to X\otimes V$ and
$\delta^{\prime}=\vartheta(\omega^{\prime})\colon V^{\prime}\to X\otimes
V^{\prime}$. Then the coaction
$\delta^{\prime\prime}=\vartheta(\omega^{\prime\prime})\colon
V^{\prime\prime}\to X\otimes V^{\prime\prime}$ corresponding to the tensor
product (3.98) coincides with

$\displaystyle V\otimes
V^{\prime}\xrightarrow{\delta\otimes\delta^{\prime}}X\otimes V\otimes X\otimes
V\xrightarrow{\sigma^{(23)}}X\otimes X\otimes V\otimes
V^{\prime}\xrightarrow{\mu_{X}\otimes\mathop{\mathrm{id}}\nolimits\otimes\mathop{\mathrm{id}}\nolimits}X\otimes
V\otimes V^{\prime}.$ (3.100)

Proof. By sequentially applying (2.39), definition of $\pi$, naturality of
$\sigma$ and the formula (2.38) we derive

$\theta\big{(}\pi\cdot(\rho\otimes\rho^{\prime})\big{)}=\theta(\pi)\cdot(\rho\otimes\rho^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}})=(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot\sigma^{(23)}\cdot(\rho\otimes\rho^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{V}\otimes\mathop{\mathrm{id}}\nolimits_{V^{\prime}})=\\\
(\mathop{\mathrm{ev}}\nolimits\otimes\mathop{\mathrm{ev}}\nolimits)\cdot(\rho\otimes\mathop{\mathrm{id}}\nolimits_{V}\otimes\rho^{\prime}\otimes\mathop{\mathrm{id}}\nolimits_{V^{\prime}})\cdot\sigma^{(23)}=(\theta(\rho)\otimes\theta(\rho^{\prime}))\cdot\sigma^{(23)}=(a\otimes
a^{\prime})\cdot\sigma^{(23)}.$ (3.101)

By using (2.39) again we obtain

$\displaystyle
a^{\prime\prime}=\theta(\rho^{\prime\prime})=\theta\big{(}\pi\cdot(\rho\otimes\rho^{\prime})\cdot\Delta_{X}\big{)}=\theta\big{(}\pi\cdot(\rho\otimes\rho^{\prime})\big{)}\cdot(\Delta_{X}\otimes\mathop{\mathrm{id}}\nolimits_{V\otimes
V^{\prime}}),$ (3.102)

Substitution of (3.101) to (3.102) yields (3.99). ∎

######  Remark 3.22.

If ${\mathbf{C}}={\mathbf{P}}$ is a closed monoidal category, then the
equivalence (3.25) induces a monoidal structure from the category
$\mathop{\mathbf{Rep}}\nolimits_{\mathbf{C}}({\mathbb{B}})$ to
$\mathop{\mathbf{Lact}}\nolimits({\mathbb{B}})$. It is claimed in [Par]
(without proof) that for arbitrary symmetric monoidal category ${\mathbf{C}}$
and a bimonoid ${\mathbb{B}}\in\mathop{\mathbf{Bimon}}\nolimits({\mathbf{C}})$
the category $\mathop{\mathbf{Lact}}\nolimits({\mathbb{B}})$ is monoidal with
respect to the monoidal product defined by the formula (3.99). Dually,
$\mathop{\mathbf{Lcoact}}\nolimits({\mathbb{B}})$ is a monoidal category with
the structure defined by (3.100). By this logic, Prop. 3.21 means that the
embeddings (3.25) and (3.27) are strict monoidal functors.

###  3.3 Translation of representations by a functor

If a functor between two relatively (co)closed monoidal categories respects
their structures in an appropriate way, then this functor induces functors
between the categories of (co)monoids, bimonoids, (co)actions and
(co)representations. We describe the functor which translates
(co)representations in details by following ideas of [S21, § 3.3] (we adapt
them to the case of relatively (co)closed monoidal categories). As above we do
not use the actions and their connection with representations. For the case of
bimonoid we study the translation of the tensor product of (co)representations
and get a monoidal structure of the corresponding induced functor.

3.3.1. Translation of internal hom. Let ${\mathbf{C}}=({\mathbf{C}},\otimes)$
and ${\mathbf{D}}=({\mathbf{D}},\odot)$ be symmetric monoidal categories. Let
${\mathbf{P}}\subset{\mathbf{C}}$ and ${\mathbf{Q}}\subset{\mathbf{D}}$ be
their full subcategories. Suppose that ${\mathbf{C}}$ and ${\mathbf{D}}$ are
relatively closed with the parametrising categories ${\mathbf{P}}$ and
${\mathbf{Q}}$ respectively.

Let $F=(F,\varphi,\phi)\colon({\mathbf{C}},\otimes)\to({\mathbf{D}},\odot)$ be
a lax monoidal functor: it consists of a functor
$F\colon{\mathbf{C}}\to{\mathbf{D}}$, a morphism $\varphi\colon
I_{\mathbf{D}}\to FI_{\mathbf{C}}$ and a natural transformation $\phi$ with
the components $\phi_{X,Y}\colon FX\odot FY\to F(X\otimes Y)$ satisfying
necessary conditions.

Suppose $F({\mathbf{P}})\subset{\mathbf{Q}}$. Then we can define the hom-
object $\mathop{\mathrm{\bf hom}}\nolimits(FV,FW)$ in ${\mathbf{D}}$ for any
$V,W\in{\mathbf{P}}$ and relate it with the image $F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}$ by a natural morphism as follows (see [Bor2, Prop.
6.4.5] for the closed case).

Let us apply the right adjunction bijection

$\displaystyle\theta^{-1}\colon\mathop{\mathrm{Hom}}\nolimits\big{(}F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}\odot
FV,FW\big{)}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\sim$}}}\mathop{\mathrm{Hom}}\nolimits\big{(}F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)},\mathop{\mathrm{\bf hom}}\nolimits(FV,FW)\big{)}$

to the composition

$\displaystyle F\big{(}\mathop{\mathrm{\bf hom}}\nolimits(V,W)\big{)}\odot
FV\xrightarrow{\phi}F\big{(}\mathop{\mathrm{\bf hom}}\nolimits(V,W)\otimes
V\big{)}\xrightarrow{F(\mathop{\mathrm{ev}}\nolimits_{V,W})}FW$ (3.103)

and denote the result by
$\Phi_{V,W}=\theta^{-1}\big{(}F(\mathop{\mathrm{ev}}\nolimits_{V,W})\cdot\phi_{\mathop{\mathrm{\bf
hom}}\nolimits(V,W),V}\big{)}$. This is a unique morphism

$\displaystyle\Phi_{V,W}\colon F\big{(}\mathop{\mathrm{\bf
hom}}\nolimits(V,W)\big{)}\to\mathop{\mathrm{\bf hom}}\nolimits(FV,FW)$
(3.104)

such that the diagram

(3.109)

commutes.

######  Proposition 3.23.

For any objects $X\in{\mathbf{C}}$, $V,W\in{\mathbf{P}}$ and morphism $f\colon
X\to\mathop{\mathrm{\bf hom}}\nolimits(V,W)$ in ${\mathbf{C}}$ we have the
formula

$\displaystyle\theta\big{(}\Phi_{V,W}\cdot
F(f)\big{)}=F\big{(}\theta(f)\big{)}\cdot\phi_{X,V}.$ (3.110)

Proof. The formula (3.110) follows from the commutativity of the diagram
(3.109) and the naturality of $\phi_{X,Y}$ in the first argument (see the
proof of [S21, Prop. 3.9]). ∎

In the closed case the naturality of $\Phi$ is due to the point (1) of [Bor2,
Prop. 6.4.5], which was claimed without a (detailed) proof. For the relative
case we deduce the naturality of $\Phi$ by using Prop. 3.23 and the naturality
of $\phi_{X,Y}$ in $Y\in{\mathbf{P}}$.