# Functors of Variance: Heuristic Naturality and Connection to Ends

David Forsman

###### Abstract

The concept of a variance on a category is introduced as a two-sided strict
factorization system. By employing variances, we define functors of variance
in a more general setting than is usually considered, thereby eliminating the
need for their domains to be product categories. Heuristic natural
transformations are defined between functors of variance, whose domains are
connected via a span. Heuristic naturality encompasses various known types of
naturality, such as diagonal, extraordinary, and twisted naturality.

We demonstrate the connection between heuristic naturality and a generalized
comma category, revealing the bijective correspondence between heuristic
natural transformations and sections of the forgetful functor from the
generalized comma category. Moreover, we introduce the concept of limit with
respect to the notion of a heuristic transformation. This is the notion of
end. It is shown that ends can be calculated via products and equalizers, and
the Fubini theorem still holds for ends.

## Introduction

The inception of category theory was largely motivated by the notion of
naturality, which can be viewed as a formalization of the concept of
canonically constructed collections of functions. For instance, the collection
of functions consisting of projections to the first component in a binary
product exemplifies a natural transformation. However, some families of maps
do not exhibit naturality in the conventional sense. Consider, for example,
the linear isomorphisms between real Hilbert spaces and their duals. The
coherence of this family of linear maps is encapsulated by twisted naturality.

Additionally, the family of evaluation maps constitutes a natural
transformation in one variable and extra-natural in another. Diagonal
naturality enables the recovery of these previously mentioned notions of
naturality. Nonetheless, the approach to diagonal naturality tends to discard
too much information about the original setting. In their paper [1], Eilenberg
and Kelly characterize when the vertical composition of extra-natural
transformations is generally well-defined. Such a theorem proves challenging
to recover from the generalized setting of dinatural transformations. Due to
practical considerations, many mathematicians have preferred to explore extra-
natural transformations over dinatural transformations in areas such as the
theory of ends. The perspective of extra-naturality permits a suitable
generalization to ends in the context of enriched categories.

In the first section, strict factorization systems are explored and, in
particular, variances as two-sided strict factorization systems. A strict
factorization system $(E,M)$ on a category $C$ is a pair of subcategories both
containing all objects of $C$, where each morphism $f$ equals to $me$ for some
unique morphisms $m$ in $M$ and $e$ in $E$. If $(M,E)$ is also a strict
factorization system, then $(E,M)$ is said to be a variance on $C$. We show
that any compatible pair of functors defined on a variance $(E,M)$ of $C$
uniquely determines a functor with $C$ as its domain. Furthermore, we define
the notion of functor of variance. A functor $C\rightarrow D$ of variance
$(E,M)$ can be characterized as a compatible pair consisting of a covariant
functor $E\rightarrow D$ and a contravariant functor $M\rightarrow D$.
Compatibility is simply the requirement that the functors behave the same on
objects and respect the factorizations of morphisms.

The second section will be dedicated to the exploration of heuristic
naturality, a central concept in our categorical setting. The notion of a span
in categories becomes an important way to relate two functors of variance with
different domains. The general theory of heuristic transformations centers
around spans, but in the special case where the domains are products of
categories, we introduce a notion of partition of arguments. With the notion
of partition of arguments, we are able to efficiently recover previous notions
of naturality like diagonal naturality, extra-naturality, and twisted
naturality. A partition of arguments can be read from a formal expression of
labeled morphisms. This leads to the name ‘heuristic naturality’.

Consider the expression

$F(x,x,x,y,z)\xrightarrow{f_{x,y,z}}G(x,z,z)$

of morphisms labeled with objects $x,y,z$ from their respective categories.
For illustrative purposes, assume that $F$ and $G$ are functors, with $F$
being contravariant at positions $2$ and $5$, $G$ being contravariant at
position $1$, and both being covariant in other positions. We define the
heuristic naturality condition for the family of morphisms
$(f_{x,y,z})_{x,y,z}$ as the commutativity of the diagram

${{F(x,x,x,y,z)}}$${{G(x,z,z)}}$${{F(x,x^{\prime},x,y,z^{\prime})}}$${{G(x,z^{\prime},z^{\prime})}}$${{F(x^{\prime},x^{\prime},x^{\prime},y^{\prime},z^{\prime})}}$${{G(x^{\prime},z^{\prime},z^{\prime})}}$$\scriptstyle{f_{x,y,z}}$$\scriptstyle{G(id,v,v)}$$\scriptstyle{F(id,s,id,id,v)}$$\scriptstyle{F(s,id,s,t,id)}$$\scriptstyle{f_{x^{\prime},y^{\prime},z^{\prime}}}$$\scriptstyle{G(s,id,id)}$

for arbitrary morphisms $s\colon x\rightarrow x^{\prime}$, $t\colon
y\rightarrow y^{\prime}$, and $v\colon z\rightarrow z^{\prime}$ in the
respective categories. The diagram is essentially uniquely constructed from
the requirement that there are two parallel paths containing the expressions
$f_{x,y,z}$ and $f_{x^{\prime},y^{\prime},z^{\prime}}$, respectively.

The third section brings our attention to the concept of comma categories,
which play a crucial role in understanding the interplay between functors and
heuristic natural transformations. A heuristic transformation can be seen as a
section to a forgetful functor from a corresponding generalized comma
category. This provides us with a new perspective on how componentwise
naturality implying naturality relates to the lifting of functors.

Lastly, we will focus on the definitions and properties of ends and coends,
which are intimately related to the notion of heuristic naturality. We
demonstrate that in this generalized setting, the Fubini theorem of ends still
holds and ends can be computed via equalizers and products when they exist.

##  1 Functors of variance

Consider a functor $F\colon C\times C\rightarrow D$ which is contravariant in
the first and covariant in the second argument. The goal is to talk about the
functor $F$ without needing the domain of $F$ to be a product category.
Instead, we specify a structure on the domain of $F$ that chooses coherently
which morphisms are covariant and contravariant. We can then define that the
functor $F$ acts covariantly on the covariant morphisms and contravariantly on
the contravariant morphisms. The choice of covariant morphisms $E$ and
contravariant morphisms $M$ will define a two-sided strict factorization
system $(E,M)$. This structure allows us to produce a theory of functors of
mixed variance.

###  1.1 Strict factorization systems and variance

Strict factorization systems were first introduced in the paper [2] by
Grandis. We use strict factorization systems to formalize the notion of
variance on a category and the notion of a functor of mixed variance.

######  Definition 1.1.

Let $C$ be a category with subcategories $E$ and $M$. The pair $(E,M)$ is
called a strict factorization system if

  * •

the subcategories $E$ and $M$ are wide, meaning they contain all the objects
of $C$;

  * •

each morphism $f$ in $C$ has a unique factorization $f=me$ where $e$ and $m$
are morphisms in $E$ and $M$, respectively.

The pair $(E,M)$ is called a variance on $C$, if $(E,M)$ and $(M,E)$ are
strict factorization systems on $C$. If $(E,M)$ is a variance on $C$, we call
the morphisms in the subcategory $E$ covariant morphisms and those in $M$
contravariant morphisms. A covariant functor $C\rightarrow D$ where $C$ and
$D$ are categories with variance is said to preserve variance, if covariant
and contravariant morphisms are mapped to covariant and contravariant
morphisms, respectively. We denote by CatV the category of small categories
equipped with variance and variance-preserving functors among them.

The category CatV is a complete category with small coproducts. Through a
direct verification, one notices that the forgetful functor
$\textbf{CatV}\rightarrow\textbf{Cat}$ strictly creates and preserves limits
and coproducts. Recall that a functor $F\colon A\rightarrow C$ is said to
strictly create the limit of a diagram $D\colon I\rightarrow A$ if each
limiting cone $\theta^{\prime}$ over $FD$ has a unique cone $\theta$ over $D$
that $F$ maps to $\theta^{\prime}$; additionally, $\theta$ is a limiting cone.
The functor $F$ is said to preserve the limit of $D$, if each limiting cone
over $D$ is taken to a limiting cone over $FD$ via $F$. If $F$
preserves/strictly creates the limit for each diagram in $A$, then $F$ is said
to preserve/strictly create limits, similarly for colimits.

Consider a category $C$ with a strict factorization system $(E,M)$. Notice
that the subcategories $E$ and $M$ are closed under inversion. Generally, $E$
and $M$ satisfy the relative right and left cancellation laws, respectively.
Since if $rs$ is a morphism in $E$, where $r$ is a morphism in $C$ and $s$ is
a morphism in $E$, then $r=me$ for some morphisms $m$ in $M$ and $e$ in $E$,
and so $rs=mes$. Thus $m$ has to be the identity, by unique decomposition.
Thus the morphism $r=e$ is in $E$. Dually, the subcategory $M$ satisfies the
relative left cancellation law.

######  Definition 1.2.

Let $C$ be a category with a variance $(E,M)$. Examine the unique
factorizations of $f\colon x\rightarrow y$ of a morphism in $C$

${x}$${f_{t}}$${f_{s}}$${y}$$\scriptstyle{f^{e}}$$\scriptstyle{f_{m}}$$\scriptstyle{f}$$\scriptstyle{f^{m}}$$\scriptstyle{f_{e}}$

where $f^{e},f_{e}$ and $f^{m},f_{m}$ are morphisms of $E$ and $M$,
respectively. We call the factorization $f=f^{m}f^{e}$ the terminating
factorization of $f$ and $f=f_{e}f_{m}$ the starting factorization of $f$ with
respect to the variance $(E,M)$ on $C$. The objects $f_{s}$ and $f_{t}$ are
called the starting and terminating objects, respectively, associated to $f$.

Remarkably, every strict factorization system on a groupoid can be viewed as a
variance. To see this, consider a groupoid $G$ equipped with a strict
factorization system $(E,M)$. Fix a morphism $g$ in $G$. Since $g^{-1}=me$ for
some morphisms $m$ in $M$ and $e$ in $E$, we have $g=e^{-1}m^{-1}$, which
yields an $(M,E)$ factorization of $g$. Since the intersection $E\cap M$ is a
discrete category, this factorization is unique.

Turning to the case of a category $C$ with a variance $(E,M)$, suppose $f$ and
$g$ are morphisms in $C$ with $f$ covariant and $g$ contravariant. In this
setting, we have the equations:

$\displaystyle f=f^{e}=f_{e},\qquad$ $\displaystyle f^{m}=id_{y},$
$\displaystyle f_{m}=id_{x},\qquad$ $\displaystyle f_{s}=x,$ $\displaystyle
f_{t}=y,$ $\displaystyle g=g^{m}=g_{m},\qquad$ $\displaystyle g^{e}=id_{x},$
$\displaystyle g_{e}=id_{y},\qquad$ $\displaystyle g_{s}=y,$ $\displaystyle
g_{t}=x.$

Consider morphisms $x\xrightarrow{f}y\xrightarrow{g}z$ in a category $C$ with
variance $(E,M)$. The morphisms $f$ and $g$ induce a commutative diagram

${x}$${f_{t}}$${(g^{e}f^{m})_{t}=(gf)_{t}}$${f_{s}}$${y}$${g_{t}}$${(g_{m}f_{e})_{s}=(gf)_{s}}$${g_{s}}$${z}$$\scriptstyle{f}$$\scriptstyle{f_{m}}$$\scriptstyle{f^{e}}$$\scriptstyle{f^{m}}$$\scriptstyle{(g^{e}f^{m})^{e}}$$\scriptstyle{(g^{e}f^{m})^{m}}$$\scriptstyle{f_{e}}$$\scriptstyle{(g_{m}f_{e})_{m}}$$\scriptstyle{g}$$\scriptstyle{g^{e}}$$\scriptstyle{g_{m}}$$\scriptstyle{g^{m}}$$\scriptstyle{(g_{m}f_{e})_{e}}$$\scriptstyle{g_{e}}$
(1)

By the unique factorizations, we immediately obtain

$\displaystyle(gf)^{m}$ $\displaystyle=g^{m}(g^{e}f^{m})^{m}$
$\displaystyle(gf)^{e}=(g^{e}f^{m})^{e}f^{e}$ (2) $\displaystyle(gf)_{m}$
$\displaystyle=(g_{m}f_{e})_{m}f_{m}$
$\displaystyle(gf)_{e}=g_{e}(g_{m}f_{e})_{e}$ (3)

Hence the codomain of $(gf)_{m}$ is the same as the codomain of
$(g_{m}f_{e})_{m}$, and so $(g_{m}f_{e})_{s}=(gf)_{s}$. The equation on the
terminating objects is obtained similarly: $(gf)_{t}=(g^{e}f^{m})_{t}$. An
important fact about the diagram (1) is that each morphism pointing to the
right is covariant and each morphism pointing downwards is contravariant.
Later we define what a functor of variance is. Such a functor maps a morphism
$f\colon x\rightarrow y$ to a morphism $F(f)\colon F(f_{s})\rightarrow
F(f_{t})$. Hence the contravariant morphisms have their direction turned and
the direction of a covariant morphism is respected.

######  Examples 1.3.

The notion of variance is prevalent in many areas of mathematics:

  * •

Every category $A$ has a covariant variance, where each morphism is covariant,
and a contravariant variance, where each morphism is defined to be
contravariant.

  * •

A product category of categories with variance has a canonical variance
structure. We define a morphism of the product category to be covariant
(contravariant) if each projection is covariant (contravariant). This
construction directly follows from the earlier observation that the forgetful
functor $\textbf{CatV}\rightarrow\textbf{Cat}$ strictly creates and preserves
limits.

  * •

Let $N$ be a monoid considered as a single-object category. A variance on $N$
is a pair of submonoids $(E,M)$ where for each element $n$ of $N$, there exist
unique elements $e,e^{\prime}\in E$ and $m,m^{\prime}\in M$ such that $n=me$
and $n=e^{\prime}m^{\prime}$. If $N$ is a group, then as previously noted for
groupoids, any strict factorization system on $N$ will be a variance and both
submonoids $E$ and $M$ will be subgroups.

    * –

Examine the set of strictly positive integers with the multiplicative
structure. Take a partitioning of the prime numbers into $P_{1},P_{2}$ and
denote the generated submonoids by $\overline{P}_{1}$ and $\overline{P}_{2}$.
By the fundamental theorem of arithmetic,
$(\overline{P}_{1},\overline{P}_{2})$ is a variance on the multiplicative
monoid of positive integers.

    * –

Let $H$ and $N$ be groups. Assume $H$ acts on $N$. Consider the associated
outer semi-direct product $H\ltimes N$. Here, $H$ and $N$ can be considered as
subgroups of the outer semi-direct product. The pair $(N,H)$ defines a
variance on $H\ltimes N$. The group $N$ is a normal subgroup of $H\ltimes N$.

    * –

The notion of variance on a group is strictly more general than internal semi-
direct products: Consider the symmetric group $S_{4}$ of permutations of four
elements. The group $S_{4}$ has non-unique subgroups of order $8$ and $3$,
respectively. Since $3$ and $8$ are coprime integers and $|S_{4}|=3\times 8$,
it follows that all pairs $(E,M)$ form a variance on $S_{4}$ where $E$ and $M$
are subgroups of order $8$ and $3$, respectively. The fact that there are
multiple subgroups of order $3$ and $8$ and the numbers are coprime implies
that neither of the subgroups $E$ and $M$ can be normal.

  * •

Let $C$ be any category. Let $J$ be a union of path components of $C$. We may
uniquely specify a variance on $C$ by setting all the morphisms whose domain
lies in $J$ to be covariant and contravariant otherwise.

  * •

Let $I$ be a set. Let $A$ be a family of categories $A_{i}$ for $i\in I$, also
called an $I$-list of categories. Let $v\colon I\rightarrow\\{0,1\\}$ be a
function. We call $v$ an index-variance on $A$ and an index $i\in I$ is said
to be covariant if $v(i)=0$ and contravariant otherwise. Consider the product
category $\prod A=\prod_{i\in I}A_{i}$. The function $v$ defines a variance on
$\prod A$ by setting $A_{i}$ to have covariant (contravariant) variance when
$i$ is a covariant (contravariant) index for each $i\in I$. Thus, the product
$\prod A$ obtains a variance from the categories $A_{i}$, for $i\in I$, via
the product. We call this variance on $\prod A$ an index-variance with respect
to $v$.

    * –

If $A_{1},\ldots,A_{n}$ are categories and $s_{1},\ldots,s_{n}\in\\{0,1\\}$,
we say that the tuple $v=(s_{1},\ldots,s_{n})$ defines an index-variance on
the category

$A_{1}\times\cdots\times A_{n}.$

  * •

Let $C$ be a category with variance $(E,M)$. A subcategory $R$ of $C$ inherits
the variance of $C$ if $R$ is closed under factoring with respect to the
variance: If $f$ is a morphism in $R$, then $f^{m},f^{e},f_{m},f_{e}$ are
morphisms in $R$. Then the restrictions of $E$ and $M$ to $R$ define a
variance on $R$.

    * –

Consider an $I$-list of categories with variance $A$. The product category
$\prod A$ has a subcategory of finitely supported morphisms. A morphism $f$ of
$\prod A$ is finitely supported if $f_{i}$ is not an identity morphism for at
most finitely many $i\in I$. The subcategory of finitely supported morphisms
of $\prod A$ inherits the variance from $\prod A$.

One important property of a variance on a category is that a functor with a
domain equipped with a variance can be reconstructed from a compatible pair of
functors defined only on the contravariant and covariant morphisms.

######  Theorem 1.4 (Decomposition theorem).

Let $C$ be a category with variance $(E,M)$. Assume that $G\colon E\rightarrow
D$ and $H\colon M\rightarrow D$ are compatible functors: The functors $G$ and
$H$ map objects the same and

$H(f^{m})G(f^{e})=G(f_{e})H(f_{m})$

for each morphism $f$ in $C$. Then there exists a unique functor $F\colon
C\rightarrow D$ extending $G$ and $H$.

###### Proof.

The uniqueness of $F$ is clear. We show existence. Define a functor $F\colon
C\rightarrow D$ mapping the objects the same as $G$. For a morphism $f$ in $C$
we define that $F(f)=H(f^{m})G(f^{e})$. Identities are taken to identities by
$F$. Left to show is the compatibility with composition. Let
$x\xrightarrow{f}y\xrightarrow{g}z$ be morphisms in $C$. Now

$\displaystyle F(gf)$ $\displaystyle=H((gf)^{m})G((gf)^{e})$
$\displaystyle=H(g^{m}(g^{e}f^{m})^{m})G((g^{e}f^{m})^{e}f^{e})$
$\displaystyle=H(g^{m})H((g^{e}f^{m})^{m})G((g^{e}f^{m})^{e})G(f^{e})$
$\displaystyle=H(g^{m})G((g^{e}f^{m})_{e})H((g^{e}f^{m})_{m})G(f^{e})$
$\displaystyle=H(g^{m})G(g^{e})H(f^{m})G(f^{e})$ $\displaystyle=F(g)F(f)$ ∎

Let $C$ be a category with a variance $(E,M)$. The subcategories $E$ and $M$
define the canonical functor $E+M\rightarrow C$. Let $D$ be the full
subcategory of the coslice category $(E+M)/\textbf{Cat}$ consisting of
functors whose generalized kernel, as defined in [3], contains the pair of
sequences of morphisms $((f^{m},f^{e}),(f_{e},f_{m}))$ for every morphism $f$
in $C$. The previous theorem demonstrates that the functor $E+M\rightarrow C$
is initial in $D$.

Now, given a pair $(A,B)$ of subcategories of $C$, we ask whether we can
define a subcategory of $(A+B)/C$ such that if the canonical functor
$A+B\rightarrow C$ is an initial object of this subcategory, then the pair
$(A,B)$ defines a variance on $C$. An affirmative answer to this question
could allow us to generalize the notion of variance from the category Cat of
small categories to a broader class of categories.

###  1.2 Functors of mixed variance

Consider the Decomposition Theorem 1.4 where a covariant functor $A\rightarrow
C$ is reconstructed from a pair of compatible covariant functors $E\rightarrow
C$ and $M\rightarrow C$ defined on a variance $(E,M)$ on the category $A$. In
this subsection, we examine the possibility that the functor $M\rightarrow C$
is instead a contravariant functor. With a notion of compatibility, we
characterize the notion of a functor of mixed variance as a compatible pair of
functors, consisting of a covariant functor $E\rightarrow C$ and a
contravariant functor $M\rightarrow C$.

######  Definition 1.5 (Functor of mixed variance).

Let $A$ and $C$ be categories and let $(E,M)$ be a variance on $A$. A functor
$A\xrightarrow[]{F}C$ of variance $(E,M)$ consists of a pair of functions from
the objects and morphisms of $A$ to the objects and morphisms of $C$,
respectively, both denoted by $F$:

  * •

Identities are preserved under $F$.

  * •

Let $f\colon x\rightarrow y$ be an morphism of $A$. Then $F(f)\colon
F(f_{s})\rightarrow F(f_{t})$, where $f_{s}$ and $f_{t}$ are the starting and
ending objects associated to $f$ with respect to the variance $(E,M)$.

Furthermore, we require the compatibility with composition: Let
$x\xrightarrow{f}y\xrightarrow{g}z$ be morphisms in $A$. The diagram

${F(f_{s})}$${F(f_{t})}$${F(gf)_{s}}$${F(gf)_{t}}$${F(g_{s})}$${F(g_{t})}$$\scriptstyle{F(f)}$$\scriptstyle{F((g^{e}f^{m})^{e})}$$\scriptstyle{F((g_{m}f_{e})_{m})}$$\scriptstyle{F((g_{m}f_{e})_{e})}$$\scriptstyle{F(gf)}$$\scriptstyle{F(g)}$$\scriptstyle{F((g^{e}f^{m})^{m})}$

commutes. We record the equations:

$\displaystyle F(gf)=F((g^{e}f^{m})^{e})F(f)F((g_{m}f_{e})_{m})$ (4)
$\displaystyle F(gf)=F((g^{e}f^{m})^{m})F(g)F((g_{m}f_{e})_{e})$

Consider a category $A$ with a variance $(E,M)$. Let
$x\xrightarrow{f}y\xrightarrow{g}z$ be morphisms in $A$. We obtain the
commutative diagram

${x}$${f_{t}}$${(gf)_{t}}$${f_{s}}$${y}$${g_{t}}$${(gf)_{s}}$${g_{s}}$${z}$$\scriptstyle{f}$$\scriptstyle{f_{m}}$$\scriptstyle{f^{e}}$$\scriptstyle{f^{m}}$$\scriptstyle{f_{e}}$$\scriptstyle{g}$$\scriptstyle{g^{e}}$$\scriptstyle{g_{m}}$$\scriptstyle{g^{m}}$$\scriptstyle{g_{e}}$

This diagram is the same as (1). Notice again that all vertical and horizontal
arrows are contravariant and covariant, respectively. Applying a functor
$F\colon A\rightarrow C$ of variance $(E,M)$, we obtain the diagram

${F(x)}$${F(f_{t})}$${F((gf)_{t})}$${F(f_{s})}$${F(y)}$${F(g_{t})}$${F((gf)_{s})}$${F(g_{s})}$${F(z)}$$\scriptstyle{F(f)}$$\scriptstyle{F(g)}$

Erasing the objects $F(x),F(y)$ and $F(z)$ from the diagram with the
associated morphisms and adding the morphism $F(gf)$ yields the diagram

${F(f_{t})}$${F((gf)_{t})}$${F(f_{s})}$${F(g_{t})}$${F((gf)_{s})}$${F(g_{s})}$$\scriptstyle{F(f)}$$\scriptstyle{F(gf)}$$\scriptstyle{F(g)}$

which is exactly the diagram for the compatibility with composition.

Similarly, as before, we obtain a decomposition theorem, but this time for
functors of mixed variance.

######  Definition 1.6 (Compatible Pair of Functors).

Given a category $A$ with variance $(E,M)$ and a category $C$, a compatible
pair of functors consists of a covariant functor $G\colon E\rightarrow C$ and
a contravariant functor $H\colon M\rightarrow C$, satisfying the following
conditions:

  * •

$G$ and $H$ map objects the same;

  * •

$G(f^{e})H(f_{m})=H(f^{m})G(f_{e})$ holds for every morphism $f$ in $A$.

######  Theorem 1.7 (Decomposition theorem of mixed variance).

Let $A$ be a category with variance $(E,M)$, and let $C$ be a category.
Consider a functor $F\colon A\rightarrow C$ of variance $(E,M)$. We assign
$F\mapsto(F|_{M}E,F|_{M})$. This assignment is bijective on compatible pairs
$(G\colon E\rightarrow C,H\colon M\rightarrow C)$.

###### Proof.

First, we show that the assignment is well defined. For this, it suffices to
show that the functor $F$ behaves covariantly on $E$, contravariantly on $M$,
and that the diagram

${F(x)}$${F(f_{t})}$${F(f_{s})}$${F(y)}$$\scriptstyle{F(f^{e})}$$\scriptstyle{F(f_{e})}$$\scriptstyle{F(f_{m})}$$\scriptstyle{F(f)}$$\scriptstyle{F(f^{m})}$

commutes for each morphism $f\colon x\rightarrow y$ in $A$. Assume first that
$x\xrightarrow{f}y\xrightarrow{g}y$ are of covariant type. Notice that
$f^{e}=f_{e}=f$ and $f^{m}=id_{y}$ and $f_{m}=id_{x}$. Thus

$\displaystyle F(gf)$
$\displaystyle=F((g^{e}f^{m})^{e})F(f)F((g_{m}f_{e})_{m})$
$\displaystyle=F(g)F(f)F(id)$ $\displaystyle=F(g)F(f).$

Assume next that $g$ and $f$ are of contravariant type. Notice that
$f^{m}=f_{m}=f$ and $f^{e}=id_{x}$ and $f_{e}=id_{y}$. Now

$\displaystyle F(gf)$
$\displaystyle=F((g^{e}f^{m})^{e})F(f)F((g_{m}f_{e})_{m})$
$\displaystyle=F(id)F(f)F(g)$ $\displaystyle=F(f)F(g)$

Let $f$ be an arbitrary morphism of $A$. Using the first of (4) we have

$\displaystyle F(f)$ $\displaystyle=F(fid_{x})$
$\displaystyle=F((f^{e}id_{x}^{m})^{e})F(id)F((f_{m}id_{x,e})_{m}$
$\displaystyle=F(f^{e})F(f_{m})$

and using the second equation of (4)

$\displaystyle F(f)$ $\displaystyle=F(id_{y}f)$
$\displaystyle=F((id_{y}^{e}f^{m})^{m})F(id_{y})F((id_{y,m}f_{e})_{e})$
$\displaystyle=F(f^{m})F(f_{e}).$

Thus the assignment $F\mapsto(F\mid_{E},F\mid_{M})$ is well defined onto
compatible pair of functors.

Fix a compatible pair $(G\colon E\rightarrow C,H\colon M\rightarrow C)$ of
functors. Define a functor $F\colon A\rightarrow C$ of variance $(E,M)$
mapping the objects the same as $G$ and $H$. For a morphism $f\colon
x\rightarrow y$ define $F(f)=G(f^{e})H(f_{m})$. Now $F(f)\colon
F(f_{s})\rightarrow F(f_{t})$. Evidently, $F$ maps identities to identities.
To show the functoriality conditions, fix morphism
$x\xrightarrow{f}y\xrightarrow{g}z$. Now

$\displaystyle F(gf)$ $\displaystyle=G((gf)^{e})H((gf)_{m})$ (The definition
of $F$) $\displaystyle=G((g^{e}f^{m})^{e}f^{e})H((g_{m}f_{e})_{m}f_{m})$
$\displaystyle(\ref{The variance factorization equations for composite})$
$\displaystyle=G((g^{e}f^{m})^{e})G(f^{e})H(f_{m})H((g_{m}f_{e})_{m})$ (The
functoriality of $G$ and $H$)
$\displaystyle=F((g^{e}f^{m})^{e})F(f)F((g_{m}f_{e})_{m})$ (The definition of
$F$)

and using the compatibility condition we attain

$\displaystyle F(gf)$ $\displaystyle=G((gf)^{e})H((gf)_{m})$ (The definition
of $F$) $\displaystyle=H((gf)^{m})G((gf)_{e})$ (Compatibility condition)
$\displaystyle=H(g^{m}(g^{e}f^{m})^{m})G(g_{e}(g_{m}f_{e})_{e})$
$\displaystyle(\ref{The variance factorization equations for composite})$
$\displaystyle=H((g^{e}f^{m})^{m})H(g^{m})G(g_{e})G((g_{m}f_{e})_{e})$ (The
functoriality of $G$ and $H$)
$\displaystyle=H((g^{e}f^{m})^{m})G(g^{e})H(g_{m})G((g_{m}f_{e})_{e})$
(Compatibility condition)
$\displaystyle=F((g^{e}f^{m})^{m})F(g)F((g_{m}f_{e})_{e})$ (The definition of
$F$).

Lastly, the functor $F$ of variance $(E,M)$ is the unique functor that the
assignment maps to $(G,H)$. This proves the claim. ∎

A compatible pair of functors defined on covariant and contravariant
morphisms, respectively, gives rise to a functor of mixed variance from the
whole domain. This makes it easier to conceptualize functors of mixed
variance.

######  Examples 1.8.

  * •

A covariant functor $C\rightarrow D$ is a functor of mixed variance with
respect to the covariant variance on $C$. Similarly for contravariant functors
from $C$.

  * •

Let $C$ be a locally small category. The hom-functor is a functor $C\times
C\rightarrow\textbf{Set}$ of index-variance $(1,0)$.

  * •

Let $C$ be a category with variance $(E,M)$. Let $F\colon C\rightarrow D$ be a
functor. Assume that $F(m)$ is invertible for each morphism $m$ in $M$. We can
define a functor $G\colon C\rightarrow D$ of variance $(E,M)$ mapping objects
the same as $F$. We set $G(e)=F(e)$ and $G(m)=F(m)^{-1}$ for all morphism $e$
and $m$ in $E$ and $M$, respectively. To show that we have a well-defined
functor of variance, it suffices to check the compatibility condition. Let
$f\colon x\rightarrow y$ be a morphism in $C$. We need to check that
$F(f^{e})F(f_{m})^{-1}=F(f^{m})^{-1}F(f_{e})$, which is equivalent to
$F(f^{m})F(f^{e})=F(f_{e})F(f_{m})$, which by functoriality is just the
equality $F(f)=F(f)$.

Essentially the same construction works the other way around too. If a functor
of mixed variance is given and all the contravariant morphisms are mapped to
invertible ones, we may turn the functor of mixed variance into a covariant
functor.

  * •

Let $G$ be a group with a variance $(E,M)$. The mapping $g\mapsto g^{e}$ is a
group homomorphism $G\rightarrow E$ if and only if $g^{e}=g_{e}$ for every
$g\in G$. This map restricts into two trivial group homomorphisms,
$E\xrightarrow{id_{E}}E$ and $M\xrightarrow{0}E$, and the compatibility
condition is exactly $g^{e}=g_{e}$ for each $g\in G$. If $E$ is a normal
subgroup of $G$, then this condition is equivalent to the elements of $E$
commuting with the elements of $M$, which forces an isomorphism $G\cong
E\times M,g\mapsto(g^{e},g^{m})$.

##  2 Heuristic naturality

The concept of a functor with variance allows us to define heuristic natural
transformations, which encompass the usual notions of naturality like
dinaturality and extraordinary naturality.

######  Definition 2.1 (Spans).

Let $A$ and $B$ be categories. A span on $A$ is a pair $(R,L)$, where $R$ is a
category and $L\colon R\rightarrow A$ is a functor. A span $(R,L)$ from $A$ to
$B$ is a span on $A\times B$ and we denote this by $A\rightarrow B$. By
$L_{i}$ we denote the functor $L$ post-composed with the projection $\pi_{i}$
for $i=1,2$.

######  Definition 2.2 (Heuristic naturality).

Let $A,B$ and $R$ be categories. Assume that $A$ and $B$ are associated with
variances. Let $F\colon A\rightarrow C$ and $G\colon B\rightarrow C$ be
functors of variance. Let $(R,L)$ be a span $A\rightarrow B$. Let $\eta$ be a
family of morphisms

$\eta_{x}\colon FL_{1}(x)\rightarrow GL_{2}(x)$

for each object $x$ of $R$. We call $\eta\colon F\xRightarrow[L]{}G$ a
heuristic transformation and an $L$-transformation, if $\eta$ satisfies the
heuristic naturality condition: Let $f\colon x\rightarrow y$ be a morphism in
$R$. Consider the factorizations of $g\coloneqq L_{1}(f)$ and $h\coloneqq
L_{2}(f)$

${L_{1}(x)}$${L_{2}(x)}$${h_{t}}$${g_{s}}$${L_{1}(y)}$${L_{2}(y)}$$\scriptstyle{g_{m}}$$\scriptstyle{g}$$\scriptstyle{h^{e}}$$\scriptstyle{h}$$\scriptstyle{h^{m}}$$\scriptstyle{g_{c}}$

The $L$-naturality with respect to $f$ is the requirement that the diagram

${F(L_{1}(x))}$${G(L_{2}(x))}$${F(g_{s})}$${G(h_{t})}$${F(L_{1}(y))}$${G(L_{2}(y))}$$\scriptstyle{\eta_{x}}$$\scriptstyle{G(h^{e})}$$\scriptstyle{F(g_{m})}$$\scriptstyle{F(g_{e})}$$\scriptstyle{\eta_{y}}$$\scriptstyle{G(h_{m})}$

commutes. The $L$-naturality of $\eta$ means that $\eta$ is $L$–natural with
respect to all morphisms $f$ in $R$.

The notions of dinaturality and extra-naturality involve the domains of the
functors to be product categories that have possibly the same category
appearing more than once in the domains. We define a relation called
partitioning of arguments that records which categories will be thought the
same.

######  Definition 2.3 (Partition of arguments).

Let $A$ be an $I$-list of categories. Let $R$ be an equivalence relation on
$I$ satisfying the compatibility condition where $iRj$ implies $A_{i}=A_{j}$
for each $i,j\in I$. The relation $R$ is called the partitioning of the
arguments on $A$. If, furthermore, $B$ is a $J$-list of categories we say that
a relation $S\colon A\rightarrow B$ is a partition of arguments from $A$ to
$B$, if $S$ is a partition of arguments of the concatenated $(I+J)$-list of
categories $AB$.111An $I$-list of categories $A$ can be thought as a functor
$I\rightarrow\textbf{CAT}$, where the set $I$ is identified with a small
discrete category. The concatenation of functors $I\rightarrow\textbf{CAT}$
and $J\rightarrow\textbf{CAT}$ is the canonical functor
$(I+J)\rightarrow\textbf{CAT}$, via the universal property of coproducts.

Let $A$ be an $I$-list of categories. Let $R$ be a partitioning of arguments
of $A$. The relation $R$ produces a span on $\prod A$. We define the category
$A_{R}$ as subcategory of $\prod A$ by choosing the morphisms $f$, called
$R$-morphisms, of $\prod A$ satisfying $f_{i}=f_{j}$ for all $(i,j)\in R$.
Similarly, a partition of arguments $S\colon A\rightarrow B$ produces a span
$\prod A\rightarrow\prod B$ which we also denote with $S$.

Consider functors $F\colon\prod A\rightarrow C$ and $G\colon\prod B\rightarrow
C$ of index-variances $s$ and $t$, respectively. Let $R\colon A\rightarrow B$
be a partition of the arguments. An $R$-transformation $\eta\colon
F\Rightarrow G$ consists of morphisms

$\eta_{x}\colon F(\pi_{1}x)\rightarrow G(\pi_{2}x)$

for $R$-objects $x$. Moreover, they must satisfy the following condition: Let
$f\colon x\rightarrow y$ be an $R$-morphism. Consider the factorizations of
$f$ in $\prod AB$:

${x}$${f_{t}}$${f_{s}}$${y}$$\scriptstyle{f^{e}}$$\scriptstyle{f_{m}}$$\scriptstyle{f}$$\scriptstyle{f^{m}}$$\scriptstyle{f_{e}}$

The heuristic naturality condition with respect to $f$ is then the
commutativity of the diagram

${F(\pi_{1}x)}$${G(\pi_{2}x)}$${F(\pi_{1}f_{s})}$${G(\pi_{2}f_{t})}$${F(\pi_{1}y)}$${G(\pi_{2}y)}$$\scriptstyle{\eta_{x}}$$\scriptstyle{G(\pi_{2}f^{e})}$$\scriptstyle{F(\pi_{1}f_{m})}$$\scriptstyle{F(\pi_{1}f_{e})}$$\scriptstyle{\eta_{y}}$$\scriptstyle{G(\pi_{2}f^{m})}$

It is worth noting that the $R$-category $AB_{R}$ is isomorphic to
$\prod_{i\in K}(AB)_{i}$, where $K$ is a set of representatives of the
equivalence relation $R$. We view the pre-ordered set $(I+J,R)$ as a category
and define a functor $\overline{AB}\colon(I+J,R)\rightarrow\textbf{Cat}$ that
maps the elements of $I+J$ to the same as $AB$ and maps all morphisms to
identities. We see that $(AB)_{R}$ is the limit of the functor $\overline{AB}$
and the inclusion $K\hookrightarrow(I+J,R)$ is an equivalence of categories.
This shows that $(AB)_{R}$ is isomorphic to the product $\prod_{i\in
K}(AB)_{i}$.

For instance, in practice we may encounter a generic expression of the type:

$F(x,y,y)\xrightarrow{f_{x,y}}G(x,x,y).$ (5)

Here we assume that $F\colon X\times Y\times Y\rightarrow C$ and $G\colon
X\times X\times Y\rightarrow C$ of variances $(1,0,1)$ and $(1,1,0)$,
respectively. We can deduce the naturality condition for morphisms $f_{x,y}$
from the variances of $F$ and $G$. Observe that the partitioning of the
arguments $R$ can be read from the variables used in the expression. We
identify those positions, where the same variable appears. Here $R$ has two
equivalence classes $\\{1,4,5\\}$ and $\\{2,3,6\\}$, since $x$ appears in
positions $1,4$ and $5$, and $y$ appears in positions $2,3$ and $6$. Fixing
morphisms $\alpha\colon x\rightarrow x^{\prime}$ and $\beta\colon y\rightarrow
y^{\prime}$, we can see that the $R$-naturality condition with respect to
$\alpha$ and $\beta$ is then the commutativity of the diagram

${{F(x,y,y)}}$${{G(x,y,y)}}$${{F(x^{\prime},y,y^{\prime})}}$${{G(x,y,y^{\prime})}}$${{F(x^{\prime},y^{\prime},y^{\prime})}}$${{G(x^{\prime},y^{\prime},y^{\prime})}}$$\scriptstyle{f_{x,y}}$$\scriptstyle{G(id_{x},id_{y},\beta)}$$\scriptstyle{F(\alpha,id_{y},\beta)}$$\scriptstyle{F(id_{x^{\prime}},\beta,id_{y^{\prime}})}$$\scriptstyle{f_{x^{\prime},y^{\prime}}}$$\scriptstyle{G(\alpha,\beta,id_{y^{\prime}})}$

Since the formal expression (5) yields the partition of arguments $R$, we may
leave $R$ implicit in our conversation about heuristic naturality, if the
generic expression is already understood. This explains the word ‘heuristic’
in the name ‘heuristic naturality’.

######  Examples 2.4.

We attain the notions of twisted, diagonal and extraordinary naturality as
special cases. Furthermore, the naturality of the family of evaluations is
obtained.

Naturality

Let $F\colon C\rightarrow D$ and $G\colon C\rightarrow D$ be covariant
functors. Let $\eta$ be a collection of morphisms $\eta_{x}\colon
Fx\rightarrow Gx$ indexed over the objects of $C$. The collection $\eta$
defines a natural transformation if and only if it defines a
$\Delta$-transformation where $\Delta\colon C\rightarrow C\times C$ is the
diagonal functor.

Twisted naturality

Let $F,G\colon C\rightarrow D$ be functors where $F$ is covariant and $G$
contravariant. Let $\eta$ be a collection of morphisms $\eta_{x}\colon
Fx\rightarrow Gx$ indexed over the objects of $C$. The $\Delta$-naturality for
$\eta$ is that the diagram

${Fx}$${Gx}$${Fy}$${Gy}$$\scriptstyle{F(f)}$$\scriptstyle{\eta_{x}}$$\scriptstyle{\eta_{y}}$$\scriptstyle{G(f)}$

commutes for any morphism $f\colon x\rightarrow y$ in $C$.

Diagonal naturality

Let $F,G\colon C\times C\rightarrow D$ be functors with variance $(1,0)$. Let
$\eta$ be a collection of morphisms $\eta_{x}\colon F(x,x)\rightarrow G(x,x)$,
then the $(\Delta,\Delta)$-naturality condition for $\eta$ (or $R$-naturality
for the appropriate partition of arguments $R$) is the commutation of the
diagram

${{F(x,x)}}$${{G(x,x)}}$${{F(y,x)}}$${{G(x,y)}}$${{F(y,y)}}$${{G(y,y)}}$$\scriptstyle{\eta_{x}}$$\scriptstyle{G(id_{x},f)}$$\scriptstyle{F(f,id_{x})}$$\scriptstyle{F(id,f)}$$\scriptstyle{\eta_{y}}$$\scriptstyle{G(f,id_{x})}$

for any morphism $f\colon x\rightarrow y$ in $C$. This is exactly the
condition for diagonal naturality. It is of interest to note that every
heuristic transformation between functors of index-variance can be thought of
as a dinatural transformation: Let $A$ be an $I$-list of categories and let
$B$ be a $J$-list of categories. Given a span $(R,L)\colon\prod
A\rightarrow\prod B$ and functors of index variance $F\colon\prod A\rightarrow
C$ and $G\colon\prod B\rightarrow C$, we define functors
$\overline{F},\overline{G}\colon R\times R\rightarrow C$ with index-variance
$(1,0)$ as follows: For each index $i\in I+J$, if $i$ is a covariant index, we
consider the functor $R\times R\xrightarrow{\pi_{2}}R\xrightarrow{\pi_{i}\circ
L}(AB)_{i}$. If $i$ is a contravariant index, we consider the functor $R\times
R\xrightarrow{\pi_{1}}R\xrightarrow{\pi_{i}\circ L}(AB)_{i}$. As $i$ runs over
the set $I+J$, the considered functors, by the universal property of the
product, yield functors $R\times R\rightarrow\prod A$ and $R\times
R\rightarrow\prod B$. Post-composing these functors with $F$ and $G$,
respectively, yields the functors of variance $\overline{F}$ and
$\overline{G}$.

Now, consider the family $\eta$ of morphisms
$\eta_{x}\colon\overline{F}(x,x)\rightarrow\overline{G}(x,x)$ for each object
$x$ of $R$. After a direct verification, we find that $\eta$ defines a
dinatural transformation $\overline{F}\Rightarrow\overline{G}$ if and only if
the family of morphisms $\eta$ defines an $L$-transformation $F\Rightarrow G$.

Extra-naturality

Let $F\colon A\times B\times B\rightarrow D$ and $G\colon A\times C\times
C\rightarrow D$ be functors of variance $(0,1,0)$. The heuristic naturality
condition for a collection $\eta$ of morphisms $\eta_{x,y,z}\colon
F(x,y,y)\rightarrow G(x,z,z),$ indexed over objects $x,y,z$ of $A,B$ and $C$,
respectively, is that the diagram

${{F(x,y,y)}}$${{G(x,z,z)}}$${{F(x,y^{\prime},y)}}$${{G(x^{\prime},z,z^{\prime})}}$${{F(x^{\prime},y^{\prime},y^{\prime})}}$${{G(x^{\prime},z^{\prime},z^{\prime})}}$$\scriptstyle{\eta_{x,y,z}}$$\scriptstyle{G(f,id_{z},h)}$$\scriptstyle{F(id_{x},g,id_{y})}$$\scriptstyle{F(f,id_{y}^{\prime},g)}$$\scriptstyle{\eta_{x^{\prime},y^{\prime},z^{\prime}}}$$\scriptstyle{G(id_{x^{\prime}},h,id_{z^{\prime}})}$

commutes for morphisms $f\colon x\rightarrow x^{\prime}$, $g\colon
y\rightarrow y^{\prime}$ and $h\colon z\rightarrow z^{\prime}$ in the
categories $A,B$ and $C$, respectively. This notion of heuristic naturality is
equivalent to extra-naturality. However, the usual condensed diagram for the
extra-naturality condition is given by the diagram

${{F(x,y,y)}}$${{G(x,z^{\prime},z^{\prime})}}$${{F(x,y^{\prime},y)}}$${{G(x^{\prime},z,z^{\prime})}}$${{F(x^{\prime},y^{\prime},y^{\prime})}}$${{G(x^{\prime},z,z)}}$$\scriptstyle{\eta_{x,y,z^{\prime}}}$$\scriptstyle{G(f,h,id_{z^{\prime}})}$$\scriptstyle{F(id_{x},g,id_{y})}$$\scriptstyle{F(f,id_{y}^{\prime},g)}$$\scriptstyle{\eta_{x^{\prime},y^{\prime},z}}$$\scriptstyle{G(id_{x^{\prime}},id_{z},h)}$

for morphisms $f,g,h$ (see [1]). This diagram does not allow an immediate
generalization, which could be one of the reasons heuristic naturality has not
been discovered before.

Naturality of evaluation

Let $C$ be a monoidal closed category with the monoidal product and internal
hom-functors $\otimes,[-,-]\colon C\times C\rightarrow C$ with variances
$(0,0)$ and $(1,0)$, respectively. Denote the counit of the adjunction as a
family of morphisms

$ev_{a,b}\colon[a,b]\otimes a\rightarrow b\text{ for objects $a,b$ in $C$}.$

From the formal expression itself, we may guess, what kind of heuristic
naturality condition $ev$ satisfies. Let $f\colon a\rightarrow a^{\prime}$ and
$g\colon b\rightarrow b^{\prime}$ be morphisms in $C$. Then the naturality
condition is that the diagram

${{[a,b]\otimes a}}$${b}$${{[a^{\prime},b]\otimes
a}}$${{[a^{\prime},b^{\prime}]\otimes
a^{\prime}}}$${b^{\prime}}$$\scriptstyle{ev_{a,b}}$$\scriptstyle{g}$$\scriptstyle{[id,g]\otimes
f}$$\scriptstyle{[f,id]\otimes id}$$\scriptstyle{ev_{a^{\prime},b^{\prime}}}$

commutes. This is the correct notion of naturality for the evaluation maps.

The family of evaluation maps of a monoidal closed category is a motivating
example for the notion of generalized extra-naturality, which is defined in
the paper [1] by Eilenberg and Kelly. In our formulation, a general extra-
natural transformation between functors of index variance $F\colon\prod
A\rightarrow C$ and $G\colon\prod B\rightarrow C$ consists of an
$R$-transformation $\eta\colon F\Rightarrow G$ where each equivalence class of
the partition of arguments $R$ has exactly two elements. Additionally, if
$iRj$ and $i\neq j$, then $i$ and $j$ differ in variance, if and only if they
are elements of the same set $I$ or $J$ in $I+J$.

##  3 Generalized comma categories

We define the generalized comma category, which is connected to the definition
of heuristic naturality. The generalized comma category gives us a conceptual
framework to view heuristic natural transformation as sections to a forgetful
functor. This is used to prove that for naturality, it suffices to check
naturality with respect to morphisms of a certain generating subgraph. This
generalizes the case where componentwise naturality implies naturality when
the domain of the functors is a finite product of categories.

######  Definition 3.1 (Generalized comma category).

Let $F\colon A\rightarrow C$ and $G\colon B\rightarrow C$ be functors with
variance. Let $(R,L)\colon A\rightarrow B$ be a span. We define the
generalized comma category $F\downarrow_{L}G$ as follows:

  * •

The objects are pairs $(x,\alpha\colon FL_{1}x\rightarrow GL_{2}x)$ where $x$
is an object of $R$ and $\alpha$ is a morphism in $C$.

  * •

A morphism $(x,\alpha)\rightarrow(y,\beta)$ consists of a morphism $f\colon
x\rightarrow y$ in $R$ making the diagram

${FL_{1}x}$${GL_{2}x}$${F(\pi_{1}g_{s})}$${G(\pi_{2}g_{t})}$${FL_{1}y}$${GL_{2}y}$$\scriptstyle{\alpha}$$\scriptstyle{G(\pi_{2}g^{e})}$$\scriptstyle{F(\pi_{1}g_{m})}$$\scriptstyle{F(\pi_{1}g_{e})}$$\scriptstyle{\beta}$$\scriptstyle{G(\pi_{2}g^{m})}$

commute, where $g\coloneqq L(f)$.

  * •

The composition of morphisms is defined through the composition in $C$.

Fix functors of variance $F\colon A\rightarrow C$ and $G\colon B\rightarrow
C$. Let $(R,L)\colon A\rightarrow B$ be a span. To see that the composition in
the comma category $F\downarrow_{L}G$ is well defined, fix morphism
$(x,\alpha)\xrightarrow{p}(y,\beta)\xrightarrow{q}(z,\gamma)$ in
$F\downarrow_{L}G$. Denote $f=L(p)$ and $g=L(q)$. Now the diagram

${F\pi_{1}x}$${G\pi_{2}x}$${F\pi_{1}f_{s}}$${G\pi_{2}f_{t}}$${F\pi_{1}(gf)_{s}}$${F\pi_{1}y}$${G\pi_{2}y}$${G\pi_{2}(gf)_{t}}$${F\pi_{1}g_{s}}$${G\pi_{2}g_{t}}$${F\pi_{1}z}$${G\pi_{2}z}$$\scriptstyle{\alpha}$$\scriptstyle{G\pi_{2}f^{e}}$$\scriptstyle{F\pi_{1}f_{m}}$$\scriptstyle{F\pi_{1}f_{e}}$$\scriptstyle{G\pi_{2}(g^{e}f^{m})^{e}}$$\scriptstyle{F\pi_{1}(g_{m}f_{e})_{m}}$$\scriptstyle{F\pi_{1}(g_{m}f_{e})_{e}}$$\scriptstyle{\beta}$$\scriptstyle{G\pi_{2}f_{m}}$$\scriptstyle{G\pi_{2}g^{e}}$$\scriptstyle{F\pi_{1}g_{e}}$$\scriptstyle{F\pi_{1}g^{m}}$$\scriptstyle{G\pi_{2}(g^{e}f^{m})^{m}}$$\scriptstyle{\gamma}$$\scriptstyle{G\pi_{2}g_{m}}$

commutes by the functoriality of $F$ and $G$ and the definition of the
morphisms $\alpha,\beta$ and $\gamma$. Thus $q\circ p$ defines a morphism
$(x,\alpha)\rightarrow(z,\gamma)$ in the comma category.

######  Examples 3.2.

The generalized comma category generalizes the notion of comma category, the
category of algebras and coalgebras for contravariant and covariant functors.
Interestingly, it contains the notion of heuristic naturality as well.

  * •

Let $F\colon A\rightarrow C$ and $G\colon B\rightarrow C$ be functors with
variance and let $(R,L)\colon A\rightarrow B$ be a span. Consider the
forgetful functor $U\colon F\downarrow_{L}G\rightarrow R$. The
$L$-transformations are in bijective correspondence with sections to $U$. An
$L$-transformation $\alpha\colon F\Rightarrow G$ produces a section via
$x\mapsto(x,\alpha_{x})$ and $[f\colon x\rightarrow
y]\mapsto[f\colon(x,\alpha_{x})\rightarrow(y,\alpha_{y})]$. The naturality
condition of $\alpha$ concerning $f$ is exactly the condition for $\alpha(f)$
to be a well-defined morphism in the comma category.

  * •

Let $F\colon A\rightarrow C$ and $G\colon B\rightarrow C$ be covariant
functors. The usual comma category $F\downarrow G$ is the same as
$F\downarrow_{Id}G$ where $Id\colon A\times B\rightarrow A\times B$ is the
identity functor.

  * •

Let $F,G\colon A\rightarrow B$ be a covariant functors. Reflect on the
diagonal functor $\Delta\colon A\rightarrow A\times A$ and the comma category

$F\downarrow_{\Delta}G.$

The objects of $F\downarrow_{\Delta}G$ could be called $F,G$-algebras. The
objects consist of pairs $(a,t\colon F(a)\rightarrow G(a))$ where $a$ is an
object of $A$ and $t$ is a morphism in $B$. A morphism
$f\colon(a,s)\rightarrow(b,t)$ is a morphism $a\xrightarrow{f}b$ in $A$ making
the diagram

${F(a)}$${F(b)}$${G(a)}$${G(b)}$$\scriptstyle{s}$$\scriptstyle{F(f)}$$\scriptstyle{t}$$\scriptstyle{G(f)}$

commute. If $F$ or $G$ is the identity functor on $A$, then
$F\downarrow_{\Delta}G$ is the category of $F$-algebras or the category
$G$-coalgebras, respectively.

Assume that $F$ and $G$ are the covariant power-set functor
$\mathcal{P}\colon\textbf{Set}\rightarrow\textbf{Pos}$ from the category of
sets to the category of posets. Each topological space can be seen as a pair
$(X,T)$, where $T\colon\mathcal{P}(X)\rightarrow\mathcal{P}(X)$ is the closure
operation, meaning it is an increasing map satisfying the conditions

$\displaystyle T(T(A))=T(A)$ $\displaystyle A\subset T(A)$ $\displaystyle
T(\emptyset)=\emptyset$ $\displaystyle T(A\cup B)=T(A)\cup T(B)$

for every $A,B\subset X$. The fixed points of $T$ define the family of closed
sets of $X$. A continuous map $f\colon(X,S)\rightarrow(Y,T)$ consists of a
function $X\rightarrow Y$ and $f(S(A))\subset T(f(A))$, which corresponds to
$2$-categorical diagram

${\mathcal{P}(X)}$${\mathcal{P}(Y)}$${\mathcal{P}(X)}$${\mathcal{P}(Y)}$$\scriptstyle{S}$$\scriptstyle{\mathcal{P}(f)}$$\scriptstyle{T}$$\scriptstyle{\mathcal{P}(f)}$${\leq}$

If the inequality is equality, then the function $f$ is continuous and closed.
Thus we see that the category of topological spaces with continuous closed
maps is the full subcategory of the closure operations in
$\mathcal{P}\downarrow_{\Delta}\mathcal{P}$.

  * •

Let $X$ be a set. Let $C$ be a locally small category. Denote with
$\overline{X}$ the functor $1\rightarrow\textbf{Set}$ from the terminal
category to the category of sets choosing the set $X$. Examine the category
$\overline{X}\downarrow_{1\times\Delta}\text{Hom}_{C}$. Here objects consists
of pairs $(d,f\colon X\rightarrow\text{Hom}_{C}(d,d))$. A morphism from
$(c,f)$ to $(d,g)$ consists of a morphism $\phi\colon c\rightarrow d$ making
the diagram

${{\text{Hom}_{C}(c,c)}}$${X}$${{\text{Hom}_{C}(c,d)}}$${{\text{Hom}_{C}(d,d)}}$$\scriptstyle{\phi_{*}}$$\scriptstyle{f}$$\scriptstyle{g}$$\scriptstyle{\phi^{*}}$

commute. This makes $\phi$ an equivariant morphism. If $X$ is a monoid, then
the full subcategory of monoid homomorphisms in
$\overline{X}\downarrow_{1\times\Delta}\text{Hom}_{C}$ defines the functor
category $[X,C]$, where the monoid $X$ is considered a single object category.

######  Theorem 3.3 (Componentwise naturality).

Let $F\colon C\rightarrow D$ be a faithful functor. Let $L\colon B\rightarrow
D$ be a functor. Let $B^{\prime}$ be a subgraph of $B$ generating $B$. Let
$S\colon B^{\prime}\rightarrow D$ be a graph morphism. Assume that
$F(S(f))=L(f)$ for each morphism $f$ in $B^{\prime}$. Then $S$ extends
uniquely to a functor $S\colon B\rightarrow D$. Furthermore, $FS=L$.

###### Proof.

The uniqueness of the extension is clear since $B^{\prime}$ generates $B$. For
existence, fix a morphism $f$ in $B$. Since $B^{\prime}$ generates $B$, there
exists a path $(f_{n},\ldots,f_{1})$ in $B^{\prime}$ whose composite in $B$ is
$f$. We define $S(f)\coloneqq S(f_{n})\cdots S(f_{1})$. By faithfulness of
$F$, $S$ is a well-defined function on morphisms. The functoriality of $S$
follows from the faithfulness of $F$ and the functoriality of $L$. It is clear
that $FS=L$. ∎

Consider a generalized comma category $F\downarrow_{L}G$ where $F\colon
A\rightarrow C$ and $G\colon B\rightarrow C$ are functors with variance and
$(R,L)\colon A\rightarrow B$ is a span. The forgetful functor $U\colon
F\downarrow_{L}G\rightarrow R$ is faithful. As an immediate consequence of
Theorem 3.3, to check the heuristic naturality of a family of morphisms
$FL_{1}x\rightarrow GL_{2}x$ for objects $x$ in $R$, it suffices to check the
naturality with respect to some class of morphisms that generate $R$.

Let $I$ and $J$ be finite sets. Let $A$ be an $I$-list of categories and $B$
be a $J$-list of categories. Fix functors of variance $F\colon\prod
A\rightarrow C$ and $G\colon\prod B\rightarrow C$. The variances need not be
index-variances. Given a partition of arguments $R\colon A\rightarrow B$ and a
family $\eta$ of morphisms $\eta_{x}\colon F(\pi_{1}x)\rightarrow G(\pi_{2}x)$
for $R$-objects $x$. To check the naturality of $\eta$, it suffices to check
the naturality of $R$-morphisms $f$, where $f$ is supported only on a single
equivalence class of $R$. This follows from Theorem 3.3, since the morphisms
that are identities up to an equivalence class of $R$ generate the rest. We
may even allow the sets $I$ and $J$ to be infinite, but then the previous
argument shows naturality with respect to morphisms supported on finitely many
equivalence classes.

##  4 Ends

We define the notion of a wedge and a cowedge along a span on the domain of a
functor of variance. As usual, a universal wedge defines an end and a
universal cowedge defines a coend. The span on the domain functions as a
choice along which the end is to be taken. The traditional definition is
obtained when the span is chosen to be the diagonal $C\rightarrow C\times C$
and the functor to be of index variance $(1,0)$ for a category $C$. We show
that the usual formula for calculating ends via limits still holds as does the
Fubini theorem.

######  Definition 4.1 ($L$-wedge).

Let $F\colon A\rightarrow C$ be a functor with variance. Let $(R,L)$ be a span
on $A$. An $L$-wedge is a pair $(c,\eta\colon\overline{c}\Rightarrow F)$,
where $c$ is an object of $C$ and $\eta$ is an $L$-transformation. A morphism
$f$ between $L$-wedges $(c,\eta)$ and $(d,\theta)$ is a morphism $f\colon
c\rightarrow d$ making the diagram

${c}$${F(Lx)}$${d}$$\scriptstyle{f}$$\scriptstyle{\eta_{x}}$$\scriptstyle{\theta_{x}}$

commute for all objects $x$ of $R$. Via the composition in $C$, we attain the
composition between morphisms of $L$-wedges. An $L$-cowedge is a pair
$(c,F\xRightarrow[L]{}\overline{c})$. We define the categorical structure of
$L$-cowedges similarly as compared to $L$-wedges.

To be explicit, an $L$-wedge $(c,\eta)$ of a functor $F\colon A\rightarrow C$
with variance consists of an object $c$ of $C$ and a collection $\eta$ of
morphisms

$(\eta_{x}\colon c\rightarrow F(Lx))_{x\in R}$

such that any morphism $f\colon x\rightarrow y$ in $R$ yields a commuting
diagram

${F(Lx)}$${c}$${F(L(f)_{t})}$${F(Ly)}$$\scriptstyle{F(L(f)^{e})}$$\scriptstyle{\eta_{x}}$$\scriptstyle{\eta_{y}}$$\scriptstyle{F(L(f)^{m})}$

######  Definition 4.2 (Ends).

Let $F\colon A\rightarrow C$ be a functor with variance and let $(R,L)$ be a
span on $A$. We call the terminal $L$-wedge of $F$ the end of $F$ over $L$ and
denote it by $(\int_{L}F,\omega\colon\overline{\int_{L}F}\xRightarrow[R]{}F)$
or by just $\int_{L}F$ and leave the universal wedge $\omega$ implicit. We
denote by $\int^{L}F$ the initial $L$-cowedge of $F$ which is also called the
coend of $F$ over $L$.

It is useful to note now that any universal wedge will define a jointly monic
family of morphisms.

######  Theorem 4.3.

Let $F\colon A\rightarrow C$ be a functor with variance. Let $(R,L)$ be a span
on $A$. Assume that the products

$\prod_{x}F(Lx)\text{ and }\prod_{f}F(L(f)_{t})$

exist, where the indexing is over objects and morphisms of $R$, respectively.
Consider the canonical morphisms

$s,t\colon\prod_{x}F(Lx)\rightarrow\prod_{f}F(L(f)_{t})$

defined by the diagram

${F(x)}$${\prod_{x}F(x)}$${\prod_{f}F(L(f)_{t})}$${F(L(f)_{t})}$${F(y)}$$\scriptstyle{F(L(f)^{e})}$$\scriptstyle{\pi_{x}}$$\scriptstyle{\pi_{y}}$$\scriptstyle{s}$$\scriptstyle{t}$$\scriptstyle{\pi_{f}}$$\scriptstyle{F(L(h)^{m})}$

where the upper part and the lower part commute for every morphism $f\colon
x\rightarrow y$ in $R$, respectively. Then the transposition

$(c,\eta\colon\overline{c}\xRightarrow[R]{}F)\mapsto(c,\eta\colon
c\rightarrow\prod_{x}F(x))$

defines an isomorphism between the category of $L$-wedges of $F$ and the
category of equalizing cones of $s$ and $t$. Especially, $\int_{L}F$ exists if
and only the equalizer of $t$ and $s$ exists, and in such a situation they
agree.

###### Proof.

It suffices to show that the transposition determines a well-defined bijection
between the $L$-wedges and equalizing cones since the functoriality on
morphisms is defined by the identity functor. Let $\eta_{x}\colon c\rightarrow
F(Lx)$ be a morphism for every object $x$ of $R$. We show that the collection
of morphisms $\eta=(\eta_{x})_{x}$ defines an $L$-wedge if and only if the
adjoint transposition $\eta^{\prime}\colon c\rightarrow\prod_{x}F(x)$
equalizes $s$ and $t$. Notice that $t\eta^{\prime}=s\eta^{\prime}$ is
equivalent to $\pi_{f}t\eta^{\prime}=\pi_{f}s\eta^{\prime}$ for every morphism
$f\colon x\rightarrow y$ in $R$, which is equivalent with
$F(L(f)^{e})\eta_{x}=F(L(f)^{m})\eta_{y}$ for every morphism $f\colon
x\rightarrow y$ in $R$. This is exactly the condition for $\eta$ to define an
$L$-wedge. ∎

In Theorem 4.3 we may weaken the assumptions a bit. It suffices that the
product

$\prod_{f}F(L(f)_{t})$

exists where indexing is over morphisms $f$ in a generating subgraph
$R^{\prime}$ of $R$. This is due to Theorem 3.3, since to check naturality it
suffices to check it only on morphisms in $R^{\prime}$.

######  Definition 4.4.

Let $A$ and $B$ be categories with variance. Let $F\colon A\times B\rightarrow
C$ be a functor of variance with respect to the product variance on $A\times
B$. We denote $F(x,y)=F_{x}(y)=F^{y}(x),F_{x}(g)=F(id_{x},g)$ and
$F^{y}(f)=F(f,id_{y})$ for objects $(x,y)$ and morphisms $(f,g)$ the product
category $A\times B$. This defines the functors of variance $F_{x}\colon
B\rightarrow C$ and $F^{y}\colon A\rightarrow C$ for all objects $x$ in $A$
and $y$ in $B$.

######  Theorem 4.5 (Parameter theorem).

Let $A$ and $B$ be categories with variance. Let $(R_{1},L_{1})$ and
$(R_{2},L_{2})$ be spans on $A$ and $B$, respectively. Let $F\colon A\times
B\rightarrow C$ be a functor with variance. Assume that
$[\int_{L_{1}}F](y)\coloneqq\int_{L_{1}}F^{y}$ exists with $\omega^{y}$ being
the universal wedge for all objects $y$ in $B$. Then $\int_{L_{1}}F$ extends
uniquely to a functor of variance $B\rightarrow C$ so that the diagram

${{[\int_{L_{1}}F](g_{s})}}$${{F(L_{1}(a),g_{s})}}$${{[\int_{L_{1}}F](g_{t})}}$${{F(L_{1}(a),g_{t})}}$$\scriptstyle{\int_{L_{1}}(F)(g)}$$\scriptstyle{\omega^{g_{s}}_{a}}$$\scriptstyle{F(id,g)}$$\scriptstyle{\omega^{g_{t}}_{a}}$

commutes for every object $a$ in $R$ and morphism $g$ in $B$.

###### Proof.

Let $g$ be a morphism in $B$. First we show that
$(F(id_{L_{1}(a)},g)\circ\omega^{g_{s}}_{a})_{a\in R}$ defines an
$L_{1}$-wedge $\overline{\int_{L_{1}}F^{g_{s}}}\Rightarrow F^{g_{t}}$. This
then shows that $\int_{L_{1}}F$ is a well-defined function on morphisms. To
show the $L_{1}$-naturality, fix a morphism $f\colon x^{\prime}\rightarrow
y^{\prime}$ in $R$. Write $L_{1}(f)$ as $h\colon x\rightarrow y$. Consider the
diagram

${{F(x,g_{s})}}$${{F(x,g_{t})}}$${\int_{L_{1}}F^{g_{s}}}$${{F(h_{t},g_{s})}}$${{F(h_{t},g_{t})}}$${{F(y,g_{s})}}$${{F(y,g_{t})}}$$\scriptstyle{F(h^{e},id_{g_{s}})}$$\scriptstyle{F(id_{x},g)}$$\scriptstyle{F(h^{e},id_{g_{t}})}$$\scriptstyle{\omega^{g_{s}}_{x^{\prime}}}$$\scriptstyle{\omega^{g_{s}}_{y^{\prime}}}$$\scriptstyle{F(id_{h_{t}},g)}$$\scriptstyle{F(h^{m},id_{g_{s}})}$$\scriptstyle{F(id_{y},g)}$$\scriptstyle{F(h^{m},id_{g_{t}})}$

which commutes, since the left side commutes by the naturality of $\omega$ and
the right side commute via the functoriality of $F$. Clearly
$\int_{L_{1}}F\colon B\rightarrow C$ maps identities to identities. To show
mixed functoriality of $\int_{L_{1}}F$, fix morphisms
$b\xrightarrow{f}b^{\prime}\xrightarrow{g}b^{\prime\prime}$ in $B$. Consider
the diagrams

${\int_{L_{1}}F(f_{s})}$${\int_{L_{1}}F(f_{t})}$${{F(L_{1}(a),f_{s})}}$${{F(L_{1}(a),f_{t})}}$${\int_{L_{1}}F((gf)_{s})}$${\int_{L_{1}}F((gf)_{t})}$${{F(L_{1}(a),(gf)_{s})}}$${{F(L_{1}(a),(gf)_{t})}}$${\int_{L_{1}}F(g_{s})}$${\int_{L_{1}}F(g_{t})}$${{F(L_{1}(a),g_{t})}}$${{F(L_{1}(a),g_{t})}}$$\scriptstyle{\int_{L_{1}}F(f)}$$\scriptstyle{\int_{L_{1}}F((g^{e}f^{m})^{e})}$$\scriptstyle{F(id,f)}$$\scriptstyle{F(id,(g^{e}f^{m})^{e})}$$\scriptstyle{\int_{L_{1}}F(((g_{m}f_{e})_{e})}$$\scriptstyle{\int_{L_{1}}F(gf)}$$\scriptstyle{\int_{L_{1}}F((g_{m}f_{e})_{m})}$$\scriptstyle{F(id,(g_{m}f_{e})_{m})}$$\scriptstyle{F(id,(g_{m}f_{e})_{e})}$$\scriptstyle{F(id,gf)}$$\scriptstyle{\int_{L_{1}}F(g)}$$\scriptstyle{\int_{L_{1}}F((g^{e}f^{m})^{m})}$$\scriptstyle{F(id,g)}$$\scriptstyle{F(id,(g^{e}f^{m})^{m})}$
(6)

where the right diagram commutes. We show that the left diagram commutes as
well. We may consider a cylindrical diagram whose top face is the left part
and the bottom face is the right part of diagrams (6), when we add the
universal wedges $\omega^{b}_{a}\colon\int_{L_{1}}F^{b}\rightarrow
F(L_{1}(a),b)$ for suitable objects $b$. The cylindric diagram is known to
commute in all other faces except for the top face. This is because of the
definition of $\int_{L_{1}}F$ and the fact that $F$ is a functor of mixed
variance. Since the morphisms $\omega^{b}_{a}$ indexed over objects $a$ in $A$
are jointly monic, for $b=(gf)_{t}$, it follows from a diagram chase that the
top face commutes as well. ∎

######  Theorem 4.6 (Limits commute, Fubini).

Let $A$ and $B$ be categories with variance. Let $(R_{1},L_{1})$ and
$(R_{2},L_{2})$ be spans on $A$ and $B$, respectively. Let $F\colon A\times
B\rightarrow C$ be a functor of variance with respect to the product variance
on $A\times B$. Denote $L=L_{1}\times L_{2}\colon R\rightarrow A\times B$
Assume that the end

$\int_{L_{1}}F^{b}$

exists with the universal wedge being
$\omega^{b}\colon\overline{\int_{L_{1}}F^{b}}\Rightarrow F^{b}$ for every
object $b$ of $B$. Consider the corresponding canonical functor
$\int_{L_{1}}F\colon B\rightarrow C$. Then the transposition

$(c,\theta\colon\overline{c}\xRightarrow[L_{2}]{}\int_{L_{1}}F)\mapsto(c,\tilde{\theta}\colon\overline{c}\xRightarrow[L]{}F),$

where $\tilde{\theta}_{x,y}=\omega^{L_{2}(y)}_{x}\circ\theta_{y}$ for objects
$(x,y)$ in $R$, defines an isomorphism between the categories of $L$-wedges of
$F$ and the $L_{2}$-wedges of $\int_{L_{1}}F$. In particular, the end
$\int_{L}F$ exists if and only if the end $\int_{L_{2}}\int_{L_{1}}F$ exists,
and if one exists, they agree.

###### Proof.

First, we show that the transposition is well-defined. Let $(c,\theta)$ be an
$L_{2}$-wedge of $\int_{L_{1}}F$. We show that $\tilde{\theta}$, which is the
collection of morphisms

$\tilde{\theta}_{x,y}\colon
c\xrightarrow{\theta_{y}}\int_{L_{1}}F^{L_{2}(y)}\xrightarrow{\omega^{L_{2}(y)}_{x}}F(L(x,y)),\text{
for objects $(x,y)$ in $R$},$

defines an $L$-wedge. Let $(p,q)\colon(a,x)\rightarrow(b,y)$ be a morphism in
$R$. Denote $(f,g)\coloneqq L(p,q)$. Consider the commutative diagram

${c}$${\int_{L_{1}}F^{L_{2}(x)}}$${{FL(a,x)}}$${\int_{L_{1}}F^{L_{2}(y)}}$${\int_{L_{1}}F^{g_{t}}}$${{F(L_{1}(a),g_{t})}}$${{FL(b,y)}}$${{F(L_{1}(b),g_{t})}}$${{F((f,g)_{t})}}$$\scriptstyle{\theta_{x}}$$\scriptstyle{\theta_{y}}$$\scriptstyle{\omega^{L_{2}(x)}_{a}}$$\scriptstyle{\int_{L_{1}}F(g^{e})}$$\scriptstyle{F(id,g^{e})}$$\scriptstyle{\omega^{L_{2}(y)}_{b}}$$\scriptstyle{\int_{L_{1}}F(g^{m})}$$\scriptstyle{\omega^{g_{t}}_{c}}$$\scriptstyle{\omega^{g_{t}}_{a}}$$\scriptstyle{F(f^{e},id)}$$\scriptstyle{F(id,g^{m})}$$\scriptstyle{F(f^{m},id)}$

The rectangles commute, because of the $L_{2}$-naturality of $\theta$, the
definition of the functor $\int_{L_{1}}F$ and the $L_{1}$-naturality of
$\omega^{g_{t}}$.

To see the bijectivity, fix an $L$-wedge $(c,\eta)$ of $F$. We construct an
$L_{2}$-wedge $(c,\theta)$ of $\int_{L_{1}}F$. We define $\theta_{y}\colon
c\rightarrow[\int_{L_{1}}F](L_{2}(y))$ to be the unique morphism making the
diagram

${\int_{L_{1}}F^{L_{2}(y)}}$${c}$${{FL(x,y)}}$$\scriptstyle{\omega^{L_{2}(y)}_{x}}$$\scriptstyle{\eta_{(x,y)}}$$\scriptstyle{\theta_{y}}$

commute for each object $x$ in $R_{1}$. Fix an object $y$ in $R_{2}$. The
morphisms $\theta_{y}$ are well defined, since $L$ naturality of $\eta$
implies that $(\eta_{x,y})_{x}$ is $L_{1}$-natural for each $y$. To see that
the family $(\theta_{y})_{y}$ is $L_{2}$-natural, fix a morphism $p\colon
b\rightarrow y$ in $R_{2}$. Denote $g=L_{2}(p)$. Let $a$ be an object of
$R_{1}$. Consider the diagram

${{F(L_{1}(a),L_{2}(b))}}$${\int_{L_{1}}F^{L_{2}(b)}}$${c}$${\int_{L_{1}}F^{g_{t}}}$${{F(L_{1}(a),g_{t})}}$${\int_{L_{1}}F^{L_{2}(y)}}$${{F(L_{1}(a),L_{2}(y))}}$$\scriptstyle{F(id,g^{e})}$$\scriptstyle{\int_{L_{1}}F(g^{e})}$$\scriptstyle{\omega_{a}^{L_{2}(b)}}$$\scriptstyle{\theta_{b}}$$\scriptstyle{\eta_{a,b}}$$\scriptstyle{\eta_{a,y}}$$\scriptstyle{\theta_{y}}$$\scriptstyle{\omega^{g_{t}}_{a}}$$\scriptstyle{\int_{L_{1}}F(g^{m})}$$\scriptstyle{\omega_{a}^{L_{2}(y)}}$$\scriptstyle{F(id,g^{m})}$

A diagram chase using the fact that the family $(\omega_{a}^{g_{t}})_{a}$ of
morphisms is jointly monic shows that the family of morphisms
$(\theta_{y})_{y}$ is $L_{2}$-natural at $p$. This shows that the
transposition defined in the theorem is bijective. The mapping of morphisms
$(c,\theta)\xrightarrow{\alpha}(d,\phi)\mapsto(c,\tilde{\theta})\xrightarrow{\alpha}(c,\tilde{\phi})$
is well defined and extends the transposition into an isomorphism of
categories. ∎

## Acknowledgements

I would like to express a heartfelt gratitude to my supervisor, Professor Van
der Linden, for their insightful feedback, and expert guidance throughout
preparing my first article. I am grateful for their mentorship.

I extend my appreciation to my colleagues in the Junior Category Theory
Seminar hosted in UCLouvain for the inspiration.

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