$\tau$-bimonads that we also introduced in [11], are a particular case of
bimonads, where the 2-cell $\lambda$ is given in terms of a 2-cell $\tau$
which is a distributive law both on the left and on the right, both with
respect to monads and comonads. In the special case of a Yang-Baxter operator
coming from a braiding one gets examples à la [26].

###### Definition 4.6

Let $({\mathcal{K}},c)$ be a 2-category with a Yang-Baxter operator $c$ and
suppose that $b\in{\mathcal{K}}$ is a 1-endo-cell endowed with a monad and
comonad structure. We call $b$ a $c$-bimonad if $c_{b,b}$ is a distributive
law both on the left and on the right, both with respect to monads and
comonads, and the following compatibilities hold:

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\begin{picture}(24.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
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\begin{picture}(24.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
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\begin{picture}(12.0,24.0)\put(0.0,12.0){\line(1,0){0.0}}
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\end{minipage}=\operatorname{Id}_{\operatorname{Id}_{A}}.$ (22)

The following observation is inspired by Benaboú:

###### Lemma 4.7

There exists a bijection between $c$-bimonads in $({\mathcal{K}},c)$ and
compatible bilax functors $1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}},c)$. It is given by
mapping any $c$-bimonad $b:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ in $({\mathcal{K}},c)$ to the bilax
functor ${\mathcal{T}}:1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ with
${\mathcal{T}}(id_{*})=b$, whose bilax structure 2-cells agree with the
respective 2-cells of the $c$-bimonad $b$.

###### Proof.

As observed by Benaboú, a lax functor from $1$ to ${\mathcal{K}}$ defines and
is defined by a monad in ${\mathcal{K}}$. Likewise, the colaxity of
${\mathcal{T}}:1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ is equivalent to
${\mathcal{T}}(id_{*})$ being a comonad. The equations (16) and (17)
correspond to the four distributive laws of $c_{b,b}$ with
$b={\mathcal{T}}(id_{*})$ and the bilaxity conditions given in (18) correspond
to conditions (22). Such a bilax functor is also a $\tau$-bimonad in the sense
of [11, Definition 7.1] (it is a monad and a comonad with a monad-morphic and
a comonad-morphic distributive law on both sides, in the sense of (16) and
(17)). ∎∎

Similarly, we clearly have:

###### Lemma 4.8

Any bilax functor $({\mathcal{T}},\nu):1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ determines a
$\nu$-bimonad ${\mathcal{T}}(id_{*})=b$.

###### Proposition 4.9

Let $({\mathcal{F}},\nu):({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ be a bilax
functor and $b\in{\mathcal{K}}$ a $c$-bimonad. Then ${\mathcal{F}}(b)$ is a
$\nu$-bimonad in ${\mathcal{K}}^{\prime}$. That is, it satisfies the last
three axioms of (22) and a variant of the first axiom shown below:

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\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(b)$}}
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\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (23)

Moreover, ${\mathcal{F}}(b)$ is a $\tau$-bimonad in the sense of [11].

###### Proof.

The claim follows by the naturality of the (co)lax structure of
${\mathcal{F}}$ with respect to the 2-cells from the (co)monad structures of
$b$. (For the three compatibilities of the (co)unit structures for
${\mathcal{F}}(b)$, alternatively, apply ${\mathcal{F}}(\eta_{b})$ and
${\mathcal{F}}(\varepsilon_{b})$ at suitable places in the last three
equations in (18) and use the naturality of the (co)lax structure of
${\mathcal{F}}$.) The last statement follows from (16) and (17). ∎∎

###### Example 4.10

We have that ${\mathcal{F}}(id_{A})$ for all $A\in{\mathcal{O}b}{\mathcal{K}}$
are $\nu$-bimonads in ${\mathcal{K}}^{\prime}$.

Given that the notion of a Yang-Baxter operator is more general than that of a
braiding, a $\nu$-bimonad generalizes the notion of a bialgebra. Moreover, we
have that the notion of a bilax functor
${\mathcal{F}}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ recovers that of
a “braided bialgebra” from [31, Definition 5.1], in the case
${\mathcal{K}}=Del({\mathcal{C}})$, where ${\mathcal{C}}$ is a monoidal
category with a single object.

###### Example 4.11

Let $({\mathcal{K}},c)$ and $({\mathcal{K}}^{\prime},d)$ be two 2-categories
with their respective Yang-Baxter operators, and let
$b:A_{0}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A_{0}$ be a $d$-bimonad in
${\mathcal{K}}^{\prime}$. We define
${\mathcal{F}}_{b}(A)=A_{0},{\mathcal{F}}_{b}(x)=b,{\mathcal{F}}_{b}(\alpha)=\operatorname{Id}_{b}$
for all objects $A$, 1-cells $x$ and 2-cells $\alpha$ in ${\mathcal{K}}$. Then
${\mathcal{F}}_{b}$ is a bilax functor with $\nu_{f,g}=d_{b,b}$ for all
1-endocells $f,g$ in ${\mathcal{K}}$. (Alternatively, instead of the Yang-
Baxter operator $d$ in ${\mathcal{K}}^{\prime}$ one could require a Yang-
Baxter operator $\nu$ on ${\mathcal{F}}_{b}$, take a $\nu$-bimonad $b$ and
obtain a bilax functor (${\mathcal{F}}_{b},\nu)$.)

The following claim is directly proved:

###### Lemma 4.12

For a comonad $d:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ and a colax transformation between
two colax functors $\chi:{\mathcal{F}}\Rightarrow{\mathcal{G}}$ it holds:

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\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(d)$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,48.0){\line(0,-1){12.0}}
\put(6.0,36.0){\line(0,-1){24.0}} \put(18.0,36.0){\line(0,-1){6.0}}
\put(18.0,30.0){\circle*{4.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}.$

Monads and comonads in ${\mathcal{K}}$ can act on other 1-cells of
${\mathcal{K}}$. That is, for example a left module over a bimonad
$b:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
is a 1-cell $x:A^{\prime}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ endowed with a left action
$\rhd:bx\Rightarrow x$ of $b$ (in [10, Definition 2.3] the axioms are
expressed in string diagrams). Note that in category theory, what we call a
$b$-(co)module is also referred to as a $b$-(co)algebra, see for example [20].

###### Proposition 4.13

Let ${\mathcal{F}},{\mathcal{G}}$ be two bilax functors and
$b:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
a $c$-bimonad in ${\mathcal{K}}$ (indeed, a monad and comonad satisfying the
two last identities in (22)).

For a colax transformation $\phi:{\mathcal{F}}\Rightarrow{\mathcal{G}}$ of
colax functors, (24) defines a left ${\mathcal{G}}(b)$-comodule structure on
$\phi(A)$.

Dually, for a lax transformation $\psi:{\mathcal{F}}\Rightarrow{\mathcal{G}}$
of lax functors, (25) defines a left ${\mathcal{G}}(b)$-module structure on
$\psi(A)$.

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---|---|---

###### Proof.

We indicate the steps of the proof for the comodule structure. Starting from
apply first the second rule in (18), then the first rule in (12), naturality
of  , naturality of $\chi$, third rule in (22) for $b$, and lastly naturality
of $\chi$.

For the counitality, starting from  apply first the fourth rule in (22), the
second rule in (12), and lastly the fourth rule in (18). ∎∎

For ${\mathcal{F}}={\mathcal{G}}$ acting on the trivial 2-category
${\mathcal{K}}=1$, and $\phi$ and $\psi$ being endo-transformations,
Proposition 4.13 recovers [10, Proposition 2.4], proved for 2-(co)monads and
their distributive laws. The latter is a 2-categorical formulation for a fact
possibly used in (braided) monoidal categories by different authors, but we
are not aware of an exact reference.

It is important to note that modules/comodules over a $c$-bimonad in
${\mathcal{K}}$ do not form a monoidal category (unless the Yang-Baxter
operator $c$ is a half-braiding).

###  4.3 Module comonads, comodule monads and relative bimonad modules

In [20] the notion of a wreath was introduced as monad in the free completion
2-category ${\rm EM}^{M}({\mathcal{K}})$ of the 2-category ${\rm
Mnd}({\mathcal{K}})$ of monads in ${\mathcal{K}}$ under the Eilenberg-Moore
construction. Dually, cowreaths are comonads in ${\rm EM}^{C}({\mathcal{K}})$,
where the latter is the analogous completion of the 2-category ${\rm
Comnd}({\mathcal{K}})$ of comonads. In [10] the first author introduced the
2-category ${\rm bEM}({\mathcal{K}})$ so that there are forgetful 2-functors
${\rm bEM}({\mathcal{K}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\rm EM}^{M}({\mathcal{K}})$ and ${\rm
bEM}({\mathcal{K}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\rm EM}^{C}({\mathcal{K}})$. Moreover,
in loc. cit. biwreaths were introduced as bimonads in ${\rm
bEM}({\mathcal{K}})$. Biwreaths as a notion integrate both wreaths and
cowreaths as well as their mixed versions: mixed wreaths and mixed cowreaths.
In particular, a biwreath also behaves like a “(co)module (co)monad” with
respect to monad-morphic or comonad-morphic distributive law in
${\mathcal{K}}$, where the highlighted notions in the 2-categorical setting
were introduced in [10].

In the present paper, similarly to the above-mentioned idea (see diagrams (67)
and (65) of loc. cit.), but now with respect to Yang-Baxter operators, we will
consider the notions that we introduce in the definition below.

For the sake of examples that we will study further below, we record that in
[10] the 2-category ${\rm Bimnd}({\mathcal{K}})$ of bimonads in
${\mathcal{K}}$ (with respect to distributive laws) was defined, so that there
are inclusion and projection 2-functors $E_{B}:{\rm
Bimnd}({\mathcal{K}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\rm bEM}({\mathcal{K}})$ and $\pi:{\rm
bEM}({\mathcal{K}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\rm Bimnd}({\mathcal{K}})$ which are
identities on 0- and 1-cells. In [11] we have considered a variation of the
2-category ${\rm Bimnd}({\mathcal{K}})$ by changing the definition of 2-cells.
It is this other version of the 2-category that we will be interested in here.
We will recall it in Example 5.2.

###### Definition 4.14

Let $b:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ be a $c$-bimonad in a 2-category
$({\mathcal{K}},c)$ with a Yang-Baxter operator. Let
$d:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
be a comonad and a right $b$-module, and $t:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ a monad and a right $b$-module. We
say that $d$ is a (right) module comonad if the left two equations below hold,
and that $t$ is a (right) comodule monad if the right two equations below
hold:

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\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\quad\begin{minipage}{24.0pt}
\begin{picture}(24.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,24.0){\line(0,1){6.0}} \put(6.0,30.0){\circle*{4.0}}
\put(18.0,12.0){\line(0,1){4.0}} \put(18.0,16.0){\line(-1,0){12.0}}
\put(18.0,16.0){\line(-3,1){12.0}} \put(6.0,12.0){\line(0,1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle t$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,24.0){\line(0,1){6.0}} \put(6.0,30.0){\circle*{4.0}}
\put(18.0,24.0){\line(0,1){6.0}} \put(18.0,30.0){\circle*{4.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle t$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

The left hand-side versions of these notions can be clearly deduced.

We continue with some simple yoga of (co)lax and bilax functors.

Consider 1-cells:  $t$$A\vphantom{A}$$\vphantom{A}A$$d$$\vphantom{}B$$x$ in
${\mathcal{K}}$, where $t$ is a monad, $d$ is a comonad and $x$ is a right
$t$-module (via a 2-cell $\lhd$) and a right $d$ comodule (via a 2-cell
$\rho$). Recall that, as we already used before, lax functors preserve monads
and colax functors preserve comonads. Given a lax functor ${\mathcal{F}}$ and
a colax functor ${\mathcal{G}}$ one has that ${\mathcal{F}}(x)$ is a right
${\mathcal{F}}(t)$-module and a right ${\mathcal{G}}(d)$-comodule with
structure 2-cells:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(30.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(30.0,36.0)(30.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(18.0,28.0)(18.0,24.0)
\put(18.0,24.0){\line(0,-1){4.0}} \put(18.0,20.0){\line(-1,0){12.0}}
\put(18.0,20.0){\line(-3,-1){12.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(30.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(t)$}}
\put(18.0,36.0){\oval(24.0,18.0)[b]} \put(18.0,27.0){\line(0,-1){3.0}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,13.0){\line(0,1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\lhd)$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(12.0,13.0){\line(1,0){12.0}} \put(12.0,23.0){\line(1,0){12.0}}
\put(35.0,23.0){\line(-1,0){12.0}} \put(35.0,13.0){\line(-1,0){12.0}}
\put(35.0,13.0){\line(0,1){10.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage},\hskip 42.67912pt\begin{minipage}{36.0pt}
\begin{picture}(36.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(x)$}}
\put(18.0,24.0){\line(0,1){4.0}} \put(18.0,28.0){\line(-1,0){12.0}}
\put(18.0,28.0){\line(-3,1){12.0}} \put(6.0,24.0){\line(0,1){12.0}}
\put(6.0,24.0){\line(0,-1){12.0}}
\qbezier(18.0,24.0)(18.0,20.0)(24.0,18.0)\qbezier(24.0,18.0)(30.0,16.0)(30.0,12.0)
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(x)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(d)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(18.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(x)$}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\rho)$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(12.0,25.0){\line(1,0){12.0}} \put(12.0,35.0){\line(1,0){12.0}}
\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(x)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{G}}(d).$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (26)

The analogous claims hold on left sides.

###### Proposition 4.15

Bilax functors
${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ with
compatible Yang-Baxter operators preserve module comonads and comodule monads.

###### Proof.

The arguments for showing that ${\mathcal{F}}$ preserves comodule monads are
analogous to proving that it maps module comonads to module comonads.
Therefore, we will restrict ourselves to the latter. Let
$b:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
be a bimonad in $({\mathcal{K}},c)$ and $x:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ a $b$-module comonad. The bilaxity of
${\mathcal{F}}$ implies that ${\mathcal{F}}(b)$ is a $d$-bimonad with an
action on the comonad ${\mathcal{F}}(x)$. Hence, we only have to show that the
first two compatibility conditions in Definition 4.14 hold. Using the
functoriality of ${\mathcal{F}}$ we have:

$\begin{minipage}{48.0pt}
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\put(12.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(36.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(b)$}}
\qbezier(12.0,60.0)(12.0,56.0)(12.0,54.0)\qbezier(12.0,54.0)(12.0,52.0)(12.0,48.0)
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\put(12.0,48.0){\line(0,-1){3.0}} \put(36.0,36.0){\oval(12.0,18.0)[t]}
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\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle d_{x,b}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
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\put(18.0,24.0){\line(0,-1){4.0}} \put(18.0,20.0){\line(-1,0){12.0}}
\put(18.0,20.0){\line(-3,-1){12.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(42.0,24.0){\line(0,-1){4.0}} \put(42.0,20.0){\line(-1,0){12.0}}
\put(42.0,20.0){\line(-3,-1){12.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(42.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(b)$}}
\put(6.0,72.0){\line(0,-1){1.0}} \put(6.0,60.0){\line(0,1){1.0}}
\put(1.0,61.0){\line(1,0){12.0}} \put(1.0,71.0){\line(1,0){12.0}}
\put(1.0,71.0){\line(0,-1){10.0}}
\put(18.0,66.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\Delta_{d})$}}
\put(23.0,71.0){\line(-1,0){12.0}} \put(23.0,61.0){\line(-1,0){12.0}}
\put(23.0,61.0){\line(0,1){10.0}} \put(25.0,71.0){\line(1,0){12.0}}
\put(25.0,61.0){\line(1,0){12.0}} \put(25.0,61.0){\line(0,1){10.0}}
\put(42.0,66.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\Delta_{b})$}}
\put(42.0,72.0){\line(0,-1){1.0}} \put(42.0,60.0){\line(0,1){1.0}}
\put(47.0,61.0){\line(-1,0){12.0}} \put(47.0,71.0){\line(-1,0){12.0}}
\put(47.0,71.0){\line(0,-1){10.0}} \put(24.0,60.0){\oval(36.0,18.0)[b]}
\put(24.0,51.0){\line(0,-1){3.0}} \put(13.0,47.0){\line(1,0){12.0}}
\put(13.0,37.0){\line(1,0){12.0}} \put(13.0,37.0){\line(0,1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(1c1)$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(24.0,37.0){\line(1,0){12.0}} \put(24.0,47.0){\line(1,0){12.0}}
\put(47.0,47.0){\line(-1,0){12.0}} \put(47.0,37.0){\line(-1,0){12.0}}
\put(47.0,37.0){\line(0,1){10.0}} \put(36.0,24.0){\oval(36.0,18.0)[t]}
\put(36.0,36.0){\line(0,-1){3.0}} \put(18.0,24.0){\line(0,-1){1.0}}
\put(18.0,12.0){\line(0,1){1.0}} \put(13.0,13.0){\line(1,0){12.0}}
\put(13.0,23.0){\line(1,0){12.0}} \put(13.0,23.0){\line(0,-1){10.0}}
\put(30.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\lhd)$}}
\put(35.0,23.0){\line(-1,0){12.0}} \put(35.0,13.0){\line(-1,0){12.0}}
\put(35.0,13.0){\line(0,1){10.0}} \put(37.0,23.0){\line(1,0){12.0}}
\put(37.0,13.0){\line(1,0){12.0}} \put(37.0,13.0){\line(0,1){10.0}}
\put(54.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\lhd)$}}
\put(54.0,24.0){\line(0,-1){1.0}} \put(54.0,12.0){\line(0,1){1.0}}
\put(59.0,13.0){\line(-1,0){12.0}} \put(59.0,23.0){\line(-1,0){12.0}}
\put(59.0,23.0){\line(0,-1){10.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(54.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(30.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(b)$}}
\put(18.0,72.0){\oval(24.0,18.0)[b]} \put(18.0,63.0){\line(0,-1){3.0}}
\put(1.0,59.0){\line(1,0){12.0}} \put(1.0,49.0){\line(1,0){12.0}}
\put(1.0,49.0){\line(0,1){10.0}} \put(18.0,60.0){\line(0,-1){1.0}}
\put(18.0,48.0){\line(0,1){1.0}} \put(12.0,49.0){\line(1,0){12.0}}
\put(12.0,59.0){\line(1,0){12.0}}
\put(30.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\Delta_{d}\Delta_{b})$}}
\put(35.0,59.0){\line(-1,0){12.0}} \put(35.0,49.0){\line(-1,0){12.0}}
\put(35.0,49.0){\line(0,1){10.0}} \put(1.0,47.0){\line(1,0){12.0}}
\put(1.0,37.0){\line(1,0){12.0}} \put(1.0,37.0){\line(0,1){10.0}}
\put(18.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(1c1)$}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(12.0,37.0){\line(1,0){12.0}} \put(12.0,47.0){\line(1,0){12.0}}
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\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,25.0){\line(0,1){10.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(12.0,25.0){\line(1,0){12.0}} \put(12.0,35.0){\line(1,0){12.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\lhd\lhd)$}}
\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
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\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
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\put(1.0,47.0){\line(1,0){12.0}} \put(1.0,37.0){\line(1,0){12.0}}
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\put(12.0,47.0){\line(1,0){12.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(\lhd)$}}
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\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
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\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
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\end{minipage}$

The compatibility of the action of ${\mathcal{F}}(b)$ with the counit of
${\mathcal{F}}(x)$ is an immediate consequence of ${\mathcal{F}}$ being lax.
∎∎

For the next property we introduce the following notion:

###### Definition 4.16

Let $b:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ be a $c$-bimonad in a 2-category
$({\mathcal{K}},c)$ with a Yang-Baxter operator and let
$t:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}A$
be a right $b$-comodule monad. A right $t$-module and a right $b$-comodule
$x:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}B$
is a right relative $t\mbox{-}b$-module if the following relation holds:

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\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle t$}}
\put(18.0,36.0){\line(0,1){4.0}} \put(18.0,40.0){\line(-1,0){12.0}}
\put(18.0,40.0){\line(-3,1){12.0}} \put(6.0,36.0){\line(0,1){12.0}}
\put(42.0,36.0){\line(0,1){4.0}} \put(42.0,40.0){\line(-1,0){12.0}}
\put(42.0,40.0){\line(-3,1){12.0}} \put(30.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle c_{b,t}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){4.0}} \put(18.0,20.0){\line(-1,0){12.0}}
\put(18.0,20.0){\line(-3,-1){12.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(36.0,24.0){\oval(12.0,18.0)[b]} \put(36.0,15.0){\line(0,-2){3.0}}
\put(36.0,15.0){\circle*{4.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle x$}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle x$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle t$}}
\put(6.0,60.0){\line(0,-1){12.0}} \put(18.0,60.0){\line(0,-1){12.0}}
\put(18.0,48.0){\line(0,-1){4.0}} \put(18.0,44.0){\line(-1,0){12.0}}
\put(18.0,44.0){\line(-3,-1){12.0}} \put(6.0,48.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,1){4.0}} \put(18.0,28.0){\line(-1,0){12.0}}
\put(18.0,28.0){\line(-3,1){12.0}} \put(6.0,24.0){\line(0,1){12.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle x$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

Morphisms of right relative $t\mbox{-}b$-modules are right $t$-linear and
right $b$-colinear 2-cells in ${\mathcal{K}}$. Analogously, one defines a left
relative $t\mbox{-}b$-module. If $t=b$ we call $x$ a relative bimonad module.

The above notions correspond to those of relative Hopf modules [5] and Hopf
modules [22] in braided monoidal categories, which in turn are
categorifications of Hopf modules introduced in [21]. Obviously, $b$ itself is
a relative bimonad module.

###### Example 4.17

Let ${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ be a bilax
functor with compatible Yang-Baxter operator. For any 1-cell
$x:A\allowbreak\mathrel{\mathop{\hbox to10.79999pt{\rightarrowfill}}\limits}B$
the 1-cell ${\mathcal{F}}(x)$ is a left relative bimonad module over
${\mathcal{F}}(id_{B})$ and a right relative bimonad module over
${\mathcal{F}}(id_{A})$. This follows from the first equation in (18) by
setting $id_{B},id_{B},id_{B},x$, respectively $x,id_{A},id_{A},id_{A}$ for
the 1-cells $g,f,h,k$.

Analogously to Proposition 4.15 we get the following result:

###### Proposition 4.18

Bilax functors
${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ with
compatible Yang-Baxter operators preserve relative bimonad modules.

Hopf bimodules, for Hopf algebras over a field, appeared in the construction
of bicovariant differential calculi over a Hopf algebra in [33]. They were
generalized in [4, Section 4.2] to the context of a braided monoidal category
${\mathcal{C}}$. For a bialgebra $B$ in ${\mathcal{C}}$ a Hopf bimodule is a
$B$-bimodule $M$ in ${\mathcal{C}}$ which is moreover a $B$-bicomodule in the
monoidal category of $B$-bimodules ${}_{B}{\mathcal{C}}_{B}$ (for the
structures on $B$ itself the regular (co)action on $B$ and the diagonal action
on tensor product of comodules are used). The latter means that both left and
right $B$-comodule structures of $M$ are left and right $B$-bimodule
morphisms, meaning that there are four conditions to be fulfilled. Together
with simultaneously $B$-linear and $B$-bicolinear morphisms Hopf bimodules
make a category denoted by ${}^{B}_{B}{\mathcal{C}}_{B}^{B}$. We mark that the
name “Hopf bimodules” is somewhat misleading, as the Hopf structure on the
bialgebra $B$ is not necessary here.

Substituting a braided monoidal category ${\mathcal{C}}$ and a bialgebra $B$
in it with a monoidal category with a Yang-Baxter operator $c$ and a
$c$-bimonad in it, we can consider the analogous category of Hopf bimodules
${}^{B}_{B}({\mathcal{C}},c)_{B}^{B}$, where ${\mathcal{C}}$ and $B$ now have
these new meanings.

###### Corollary 4.19

Let ${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ be a bilax
functor with compatible Yang-Baxter operator. The functors (20) for every
$A\in{\mathcal{O}b}{\mathcal{K}}$ factor through the category of Hopf
bimodules over the $d$-bimonads ${\mathcal{F}}(id_{A})$ in
$({\mathcal{K}}^{\prime}({\mathcal{F}}(A),{\mathcal{F}}(A)),d)$.

###### Proof.

For any 1-endocell $x:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A$ we should check that
${\mathcal{F}}(x)$ satisfies the four relations. Two of them are satisfied by
Example 4.17, which mean that the left coaction is left linear and that right
coaction is right linear. The other two, meaning the two mixed versions of
compatibilities, one gets by setting $x=f$, respectively $x=h$, and the
resting three 1-cells to be identities, in the first equation in (18).

To check the claim on morphisms, observe that for any 2-cell
$\alpha:x\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}y$ in ${\mathcal{K}}$, i.e. morphism in
${\mathcal{K}}(A,A)$, ${\mathcal{F}}(\alpha)$ is
${\mathcal{F}}(id_{A})$-(co)linear by the naturality of the (co)lax structure
of ${\mathcal{F}}$. ∎∎

We record here some direct consequences for a bilax functor
$({\mathcal{F}},\nu)$ from the first equation in (18). For simplicity, we may
consider ${\mathcal{F}}:({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{K}}^{\prime},d)$ to have a
compatible Yang-Baxter operator. By (26) note that ${\mathcal{F}}(x)$ is a
bi(co)module over ${\mathcal{F}}(id)$ for any 1-cell $x$ (acting among
suitable 0-cells). Then we may write:

$\begin{minipage}{48.0pt} \begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}} \put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}} \put(36.0,36.0){\oval(12.0,18.0)[t]} \put(36.0,48.0){\line(0,-1){3.0}} \put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}} \put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{1,f}$}} \put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}} \put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}} \put(6.0,24.0){\line(0,-1){4.0}} \put(6.0,20.0){\line(1,0){12.0}} \put(6.0,20.0){\line(3,-1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}} \put(30.0,24.0){\line(0,-1){4.0}} \put(30.0,20.0){\line(1,0){12.0}} \put(30.0,20.0){\line(3,-1){12.0}} \put(42.0,24.0){\line(0,-1){12.0}} \put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}} \put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(k)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}=\begin{minipage}{36.0pt} \begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}} \qbezier(6.0,48.0)(6.0,44.0)(12.0,42.0)\qbezier(12.0,42.0)(18.0,40.0)(18.0,36.0) \put(30.0,48.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){4.0}} \put(18.0,32.0){\line(1,0){12.0}} \put(18.0,32.0){\line(3,-1){12.0}} \put(30.0,36.0){\line(0,-1){12.0}} \put(30.0,12.0){\oval(24.0,18.0)[t]} \put(30.0,24.0){\line(0,-1){3.0}} \put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}} \put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(k)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}$ (27) |  |  $\begin{minipage}{48.0pt} \begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}} \put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(k)$}} \put(6.0,36.0){\line(0,1){4.0}} \put(6.0,40.0){\line(1,0){12.0}} \put(6.0,40.0){\line(3,1){12.0}} \put(18.0,36.0){\line(0,1){12.0}} \put(30.0,36.0){\line(0,1){4.0}} \put(30.0,40.0){\line(1,0){12.0}} \put(30.0,40.0){\line(3,1){12.0}} \put(42.0,36.0){\line(0,1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}} \put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{f,1}$}} \put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}} \put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}} \put(36.0,24.0){\oval(12.0,18.0)[b]} \put(36.0,15.0){\line(0,-2){3.0}} \put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}=\begin{minipage}{24.0pt} \begin{picture}(24.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(f)$}} \put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(k)$}} \put(30.0,48.0){\oval(24.0,18.0)[b]} \put(30.0,39.0){\line(0,-1){3.0}} \put(18.0,24.0){\line(0,1){4.0}} \put(18.0,28.0){\line(1,0){12.0}} \put(18.0,28.0){\line(3,1){12.0}} \put(30.0,24.0){\line(0,1){12.0}} \qbezier(18.0,24.0)(18.0,20.0)(12.0,18.0)\qbezier(12.0,18.0)(6.0,16.0)(6.0,12.0) \put(30.0,24.0){\line(0,-1){12.0}} \put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}$ (28)
---|---|---

$\begin{minipage}{48.0pt} \begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}} \put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}} \put(36.0,36.0){\oval(12.0,18.0)[t]} \put(36.0,48.0){\line(0,-1){3.0}} \put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}} \put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{h,1}$}} \put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}} \put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){4.0}} \put(18.0,20.0){\line(-1,0){12.0}} \put(18.0,20.0){\line(-3,-1){12.0}} \put(6.0,24.0){\line(0,-1){12.0}} \put(42.0,24.0){\line(0,-1){4.0}} \put(42.0,20.0){\line(-1,0){12.0}} \put(42.0,20.0){\line(-3,-1){12.0}} \put(30.0,24.0){\line(0,-1){12.0}} \put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(g)$}} \put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(h)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}=\begin{minipage}{36.0pt} \begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}} \put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(18.0,48.0){\line(0,-1){12.0}} \qbezier(42.0,48.0)(42.0,44.0)(36.0,42.0)\qbezier(36.0,42.0)(30.0,40.0)(30.0,36.0) \put(30.0,36.0){\line(0,-1){4.0}} \put(30.0,32.0){\line(-1,0){12.0}} \put(30.0,32.0){\line(-3,-1){12.0}} \put(18.0,36.0){\line(0,-1){12.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]} \put(18.0,24.0){\line(0,-1){3.0}} \put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(g)$}} \put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(h)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage},$ (29) |  |  $\begin{minipage}{48.0pt} \begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(g)$}} \put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(h)$}} \put(18.0,36.0){\line(0,1){4.0}} \put(18.0,40.0){\line(-1,0){12.0}} \put(18.0,40.0){\line(-3,1){12.0}} \put(6.0,36.0){\line(0,1){12.0}} \put(42.0,36.0){\line(0,1){4.0}} \put(42.0,40.0){\line(-1,0){12.0}} \put(42.0,40.0){\line(-3,1){12.0}} \put(30.0,36.0){\line(0,1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}} \put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{1,h}$}} \put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}} \put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}} \put(36.0,24.0){\oval(12.0,18.0)[b]} \put(36.0,15.0){\line(0,-2){3.0}} \put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}} \put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}=\begin{minipage}{48.0pt} \begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}} \put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(g)$}} \put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(h)$}} \put(18.0,48.0){\oval(24.0,18.0)[b]} \put(18.0,39.0){\line(0,-1){3.0}} \put(30.0,24.0){\line(0,1){4.0}} \put(30.0,28.0){\line(-1,0){12.0}} \put(30.0,28.0){\line(-3,1){12.0}} \put(18.0,24.0){\line(0,1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}} \qbezier(30.0,24.0)(30.0,20.0)(36.0,18.0)\qbezier(36.0,18.0)(42.0,16.0)(42.0,12.0) \put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}} \put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(id)$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt} \end{minipage}$ (30)
---|---|---

for 1-cells $A\stackrel{{\scriptstyle k}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
h}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
f}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B\stackrel{{\scriptstyle
g}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}C$.

##  5 2-category of bilax functors

In this section we introduce the rest of the ingredients to construct a
2-category of bilax functors.

###  5.1 Bilax natural transformations

Among bilax functors we introduce bilax natural transformations. Recall that
for a lax transformation $\psi$ and a colax transformation $\phi$, both acting
between (bilax) functors ${\mathcal{F}}\Rightarrow{\mathcal{F}}^{\prime}$, for
every 1-cell $f:A\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B$ there are 2-cells

$\psi_{f}:{\mathcal{F}}^{\prime}_{A,B}(f)\circ\psi(A)\Rightarrow\psi(B)\circ{\mathcal{F}}_{A,B}(f)$

and

$\phi_{f}:\phi(B)\circ{\mathcal{F}}_{A,B}(f)\Rightarrow{\mathcal{F}}^{\prime}_{A,B}(f)\circ\phi(A)$

natural in $f$. For the sake of the following definition we introduce the
notation:

$\lambda_{xy,z}:=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(xy)$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(z)$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{y,z}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(xz)$}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(y)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (31)

for a bilax functor ${\mathcal{F}}$ and 1-cells $A\stackrel{{\scriptstyle
z}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}A\stackrel{{\scriptstyle
y}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}A\stackrel{{\scriptstyle
x}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B$. Then the first equation in (18)
can be expressed also as:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gf)$}}
\put(24.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(hk)$}}
\qbezier(0.0,48.0)(0.0,44.0)(3.0,42.0)\qbezier(3.0,42.0)(6.0,40.0)(6.0,36.0)
\put(24.0,36.0){\oval(12.0,18.0)[t]} \put(24.0,48.0){\line(0,-1){3.0}}
\put(6.0,36.0){\line(0,-1){1.0}} \put(6.0,24.0){\line(0,1){1.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{gf,h}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(30.0,36.0){\line(0,-1){12.0}}
\qbezier(6.0,24.0)(6.0,20.0)(3.0,18.0)\qbezier(3.0,18.0)(0.0,16.0)(0.0,12.0)
\put(24.0,24.0){\oval(12.0,18.0)[b]} \put(24.0,15.0){\line(0,-2){3.0}}
\put(0.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}}
\put(24.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{60.0pt}
\begin{picture}(60.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gf)$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(hk)$}}
\put(18.0,48.0){\oval(24.0,18.0)[b]} \put(18.0,39.0){\line(0,-1){3.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,25.0){\line(0,1){10.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(12.0,25.0){\line(1,0){12.0}}
\put(12.0,35.0){\line(1,0){12.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle{\mathcal{F}}(1c1)$}}
\put(35.0,35.0){\line(-1,0){12.0}} \put(35.0,25.0){\line(-1,0){12.0}}
\put(35.0,25.0){\line(0,1){10.0}} \put(18.0,12.0){\oval(24.0,18.0)[t]}
\put(18.0,24.0){\line(0,-1){3.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(gh)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(fk)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (32)

Observe that by (19) and the rules of the (co)lax structures of
${\mathcal{F}}$, one has:

$\begin{minipage}{24.0pt}
\begin{picture}(24.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,1){6.0}}
\put(18.0,42.0){\circle{5.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{x\hskip
1.13791ptid,id}$}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(23.0,25.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(0,-1){10.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){6.0}}
\put(18.0,18.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(x)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\operatorname{Id}_{{\mathcal{F}}(x)}.$ (33)

###### Definition 5.1

A bilax natural transformation
$\chi:{\mathcal{F}}\Rightarrow{\mathcal{F}}^{\prime}$ between bilax functors
is a pair $(\psi,\phi)$ consisting of a lax natural transformation $\psi$ of
lax functors and a colax natural transformation $\phi$ of colax functors,
which agree on the 1-cell components, i.e. $\psi(A)=\phi(A):=\chi(A)$ for
every $A\in{\mathcal{O}b}{\mathcal{K}}$, and whose 2-cell components are
related through the relation:

$\begin{minipage}{48.0pt}
\begin{picture}(48.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(-6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(xy)$}}
\put(18.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(42.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(z)$}}
\qbezier(0.0,72.0)(0.0,68.0)(3.0,66.0)\qbezier(3.0,66.0)(6.0,64.0)(6.0,60.0)
\put(18.0,72.0){\line(0,-1){12.0}}
\qbezier(42.0,72.0)(42.0,64.0)(36.0,60.0)\qbezier(36.0,60.0)(30.0,56.0)(30.0,48.0)
\put(6.0,60.0){\line(0,-1){1.0}} \put(6.0,48.0){\line(0,1){1.0}}
\put(1.0,49.0){\line(1,0){12.0}} \put(1.0,59.0){\line(1,0){12.0}}
\put(1.0,59.0){\line(0,-1){10.0}}
\put(18.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{xy}$}}
\put(18.0,60.0){\line(0,-1){1.0}} \put(18.0,48.0){\line(0,1){1.0}}
\put(23.0,49.0){\line(-1,0){12.0}} \put(23.0,59.0){\line(-1,0){12.0}}
\put(23.0,59.0){\line(0,-1){10.0}} \put(6.0,48.0){\line(0,-1){12.0}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(13.0,37.0){\line(1,0){12.0}} \put(13.0,47.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{xy,z}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{xz}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}}
\qbezier(30.0,36.0)(30.0,28.0)(36.0,24.0)\qbezier(36.0,24.0)(42.0,20.0)(42.0,12.0)
\qbezier(6.0,24.0)(6.0,20.0)(3.0,18.0)\qbezier(3.0,18.0)(0.0,16.0)(0.0,12.0)
\put(18.0,24.0){\line(0,-1){12.0}}
\put(-6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(xz)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(y)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(xy)$}}
\put(30.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(54.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(z)$}}
\qbezier(0.0,72.0)(0.0,64.0)(9.0,60.0)\qbezier(9.0,60.0)(18.0,56.0)(18.0,48.0)
\put(30.0,72.0){\line(0,-1){12.0}}
\qbezier(54.0,72.0)(54.0,68.0)(48.0,66.0)\qbezier(48.0,66.0)(42.0,64.0)(42.0,60.0)
\put(30.0,60.0){\line(0,-1){1.0}} \put(30.0,48.0){\line(0,1){1.0}}
\put(25.0,49.0){\line(1,0){12.0}} \put(25.0,59.0){\line(1,0){12.0}}
\put(25.0,59.0){\line(0,-1){10.0}}
\put(42.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{z}$}}
\put(42.0,60.0){\line(0,-1){1.0}} \put(42.0,48.0){\line(0,1){1.0}}
\put(47.0,49.0){\line(-1,0){12.0}} \put(47.0,59.0){\line(-1,0){12.0}}
\put(47.0,59.0){\line(0,-1){10.0}} \put(18.0,48.0){\line(0,-1){1.0}}
\put(18.0,36.0){\line(0,1){1.0}} \put(13.0,37.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(1,0){12.0}} \put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda^{\prime}_{xy,z}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(42.0,48.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,28.0)(9.0,24.0)\qbezier(9.0,24.0)(0.0,20.0)(0.0,12.0)
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(25.0,25.0){\line(1,0){12.0}} \put(25.0,35.0){\line(1,0){12.0}}
\put(25.0,35.0){\line(0,-1){10.0}}
\put(42.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{y}$}}
\put(42.0,36.0){\line(0,-1){1.0}} \put(42.0,24.0){\line(0,1){1.0}}
\put(47.0,25.0){\line(-1,0){12.0}} \put(47.0,35.0){\line(-1,0){12.0}}
\put(47.0,35.0){\line(0,-1){10.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\qbezier(42.0,24.0)(42.0,20.0)(48.0,18.0)\qbezier(48.0,18.0)(54.0,16.0)(54.0,12.0)
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(xz)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(54.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}(y)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (34)

for composable 1-cells: $A\stackrel{{\scriptstyle
z}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}A\stackrel{{\scriptstyle
y}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}A\stackrel{{\scriptstyle
x}}{{\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}}}B$. We will denote it shortly as a
triple $(\chi,\psi,\phi)$.

In particular, if $y=z=id_{A}$ and one applies the unity of the lax structure
of ${\mathcal{F}}$ on the top right in (34), and the counity of the colax
structure of ${\mathcal{F}}$ on the bottom right, one obtains (by (33) and
Proposition 4.13):

$\begin{minipage}{24.0pt}
\begin{picture}(24.0,72.0)\put(0.0,60.0){\line(1,0){0.0}}
\put(-6.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(18.0,64.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\qbezier(0.0,60.0)(0.0,56.0)(3.0,54.0)\qbezier(3.0,54.0)(6.0,52.0)(6.0,48.0)
\put(18.0,60.0){\line(0,-1){12.0}} \put(6.0,48.0){\line(0,-1){1.0}}
\put(6.0,36.0){\line(0,1){1.0}} \put(1.0,37.0){\line(1,0){12.0}}
\put(1.0,47.0){\line(1,0){12.0}} \put(1.0,47.0){\line(0,-1){10.0}}
\put(18.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{x}$}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(23.0,37.0){\line(-1,0){12.0}} \put(23.0,47.0){\line(-1,0){12.0}}
\put(23.0,47.0){\line(0,-1){10.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{x}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}}
\qbezier(6.0,24.0)(6.0,20.0)(3.0,18.0)\qbezier(3.0,18.0)(0.0,16.0)(0.0,12.0)
\put(18.0,24.0){\line(0,-1){12.0}}
\put(-6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,1){4.0}}
\put(18.0,40.0){\line(1,0){12.0}} \put(18.0,40.0){\line(3,1){12.0}}
\put(30.0,36.0){\line(0,1){12.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda^{\prime}_{x\hskip
1.13791ptid,id}$}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(23.0,25.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(0,-1){10.0}}
\put(30.0,36.0){\line(0,-1){12.0}} \put(6.0,24.0){\line(0,-1){12.0}}
\put(18.0,24.0){\line(0,-1){4.0}} \put(18.0,20.0){\line(1,0){12.0}}
\put(18.0,20.0){\line(3,-1){12.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,36.0){\line(0,1){4.0}} \put(30.0,40.0){\line(1,0){12.0}}
\put(30.0,40.0){\line(3,1){12.0}} \put(42.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu_{id,id}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}}
\put(30.0,24.0){\line(0,-1){4.0}} \put(30.0,20.0){\line(1,0){12.0}}
\put(30.0,20.0){\line(3,-1){12.0}} \put(42.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle{\mathcal{F}}^{\prime}(x)$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi(A)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (35)

which we will call the Yetter-Drinfel‘d condition on the bilax natural
transformation $\chi$. Observe that the left module and comodule structures of
$\chi(A)$ above are over ${\mathcal{F}}^{\prime}(id_{A})$. The relation (34)
we will call strong Yetter-Drinfel‘d condition on $\chi$, and the 1-cells
$\chi(A)$ in ${\mathcal{K}}^{\prime}$ we will call strong Yetter-Drinfel‘d
modules (over ${\mathcal{F}}^{\prime}(id_{A})$).

###### Example 5.2

Let ${\mathcal{K}}$ and ${\mathcal{K}}^{\prime}$ be 2-categories with Yang-
Baxter operators $c$ and $d$, respectively. We fix two $d$-bimonads
$b_{0}:A_{0}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A_{0}$ and
$b_{1}:A_{1}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}A_{1}$ in ${\mathcal{K}}^{\prime}$ and
define bilax functors
${\mathcal{F}}_{0},{\mathcal{F}}_{1}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ by
${\mathcal{F}}_{0}:={\mathcal{F}}_{b_{0}}$ and
${\mathcal{F}}_{1}:={\mathcal{F}}_{b_{1}}$ with $\nu_{i}=d$ for $i=1,2$, as in
Example 4.11.

A bilax natural transformation $(\chi,\psi,\phi)$ between bilax functors
$\chi:{\mathcal{F}}_{0}\Rightarrow{\mathcal{F}}_{1}$ consists of a family of
2-cells

$\psi_{A}:b_{1}\chi(A)\Rightarrow\chi(A)b_{0}\quad\text{and}\quad\phi_{A}:\chi(A)b_{0}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}b_{1}\chi(A),$

indexed by $A\in{\mathcal{O}b}{\mathcal{K}}$, where $\psi_{A}$ (respectively
$\phi_{A}$) are distributive laws with respect to the monad (resp. comonad)
structures of $b_{0},b_{1}$ (by (12) and its vertical dual), and it holds:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(6.0,48.0){\line(0,-1){1.0}} \put(6.0,36.0){\line(0,1){1.0}}
\put(1.0,37.0){\line(1,0){12.0}} \put(1.0,47.0){\line(1,0){12.0}}
\put(1.0,47.0){\line(0,-1){10.0}}
\put(18.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{A}$}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(23.0,37.0){\line(-1,0){12.0}} \put(23.0,47.0){\line(-1,0){12.0}}
\put(23.0,47.0){\line(0,-1){10.0}} \put(30.0,48.0){\line(0,-1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{0}$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){1.0}}
\put(6.0,12.0){\line(0,1){1.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,23.0){\line(0,-1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{A}$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(23.0,13.0){\line(-1,0){12.0}} \put(23.0,23.0){\line(-1,0){12.0}}
\put(23.0,23.0){\line(0,-1){10.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(6.0,48.0){\line(0,-1){12.0}} \put(18.0,48.0){\line(0,-1){1.0}}
\put(18.0,36.0){\line(0,1){1.0}} \put(13.0,37.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(1,0){12.0}} \put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{A}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{1}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(30.0,36.0){\line(0,-1){12.0}}
\put(6.0,24.0){\line(0,-1){12.0}} \put(18.0,24.0){\line(0,-1){1.0}}
\put(18.0,12.0){\line(0,1){1.0}} \put(13.0,13.0){\line(1,0){12.0}}
\put(13.0,23.0){\line(1,0){12.0}} \put(13.0,23.0){\line(0,-1){10.0}}
\put(30.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{A}$}}
\put(30.0,24.0){\line(0,-1){1.0}} \put(30.0,12.0){\line(0,1){1.0}}
\put(35.0,13.0){\line(-1,0){12.0}} \put(35.0,23.0){\line(-1,0){12.0}}
\put(35.0,23.0){\line(0,-1){10.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0},$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\quad\text{and consequently}\quad\begin{minipage}{24.0pt}
\begin{picture}(24.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(6.0,36.0){\line(0,-1){1.0}} \put(6.0,24.0){\line(0,1){1.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{A}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
\put(23.0,35.0){\line(0,-1){10.0}} \put(6.0,24.0){\line(0,-1){1.0}}
\put(6.0,12.0){\line(0,1){1.0}} \put(1.0,13.0){\line(1,0){12.0}}
\put(1.0,23.0){\line(1,0){12.0}} \put(1.0,23.0){\line(0,-1){10.0}}
\put(18.0,18.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{A}$}}
\put(18.0,24.0){\line(0,-1){1.0}} \put(18.0,12.0){\line(0,1){1.0}}
\put(23.0,13.0){\line(-1,0){12.0}} \put(23.0,23.0){\line(-1,0){12.0}}
\put(23.0,23.0){\line(0,-1){10.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,36.0){\line(0,1){4.0}} \put(30.0,40.0){\line(1,0){12.0}}
\put(30.0,40.0){\line(3,1){12.0}} \put(42.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}} \put(18.0,36.0){\line(0,-1){1.0}}
\put(18.0,24.0){\line(0,1){1.0}} \put(13.0,25.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(1,0){12.0}} \put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle d$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}}
\put(30.0,24.0){\line(0,-1){4.0}} \put(30.0,20.0){\line(1,0){12.0}}
\put(30.0,20.0){\line(3,-1){12.0}} \put(42.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle\chi_{A}.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (36)

Note that $\lambda_{0}$ and $\lambda_{1}$ have the form:

$\lambda_{0}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle d$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{0}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\quad\text{and}\qquad\lambda_{1}=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle d$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle b_{1}$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

and that the third equation in (12) is now trivial. For every
$A\in{\mathcal{O}b}{\mathcal{K}}$, the triple $(\chi(A),\psi_{A},\phi_{A})$ is
a 1-cell in the 2-category ${\rm Bimnd}({\mathcal{K}}^{\prime})$ from [11,
Section 7], which we mentioned at the beginning of Subsection 4.3. The 0-cells
of ${\rm Bimnd}({\mathcal{K}})$ are bimonads in ${\mathcal{K}}$ defined via a
distributive law 2-cell $\lambda$, 1-cells are triples $(F,\psi,\phi)$ where
$(F,\psi)$ is a 1-cell in ${\rm Mnd}({\mathcal{K}})$ and $(F,\phi)$ is a
1-cell in ${\rm Comnd}({\mathcal{K}})$ with a compatibility condition between
$\psi,\phi$ and $\lambda$ (as in (36) on the left), and a 2-cell is a single
2-cell $\zeta$ in ${\mathcal{K}}$ which is simultaneously a 2-cell in ${\rm
Mnd}({\mathcal{K}})$ and in ${\rm Comnd}({\mathcal{K}})$.

###### Example 5.3

If ${\mathcal{K}}=Del({\mathcal{C}})$ in the above Example is induced by a
braided monoidal category ${\mathcal{C}}$, the above bilax natural
transformation
$\chi:{\mathcal{F}}_{0}\Rightarrow{\mathcal{F}}_{1}:Del({\mathcal{C}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ is precisely a
single 1-cell in ${\rm Bimnd}(Del({\mathcal{C}}))$.

###### Example 5.4

By Lemma 4.8 actually any two bilax functors
$({\mathcal{T}}_{0},\nu),({\mathcal{T}}_{1},\nu^{\prime}):1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ determine two bimonads in
${\mathcal{K}}$: ${\mathcal{T}}_{0}$ yields a $\nu$-bimonad
$b_{0}:={\mathcal{T}}(id_{*})$ on $A={\mathcal{T}}(*)$ and ${\mathcal{T}}_{1}$
a $\nu^{\prime}$-bimonad $b_{1}:={\mathcal{T}}_{1}(id_{*})$ on
$A^{\prime}={\mathcal{T}}^{\prime}(*)$. Then analogously as in Example 5.3,
any bilax natural transformation
$\chi:{\mathcal{T}}_{0}\Rightarrow{\mathcal{T}}_{1}:1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ is precisely a
single 1-cell in ${\rm Bimnd}({\mathcal{K}})$.

###### Example 5.5

Consider a bilax transformation
$\chi:({\mathcal{T}},\nu)\Rightarrow({\mathcal{T}}^{\prime},\nu^{\prime}):1\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}$ from the trivial
2-category to a 2-category ${\mathcal{K}}$. Let $B:={\mathcal{T}}(id_{*})$ be
the $\nu$-bimonad on ${\mathcal{A}}={\mathcal{T}}(*)$ and
$B^{\prime}:={\mathcal{T}}^{\prime}(id_{*})$ the $\nu^{\prime}$-bimonad on
${\mathcal{T}}^{\prime}(*)$ as in Lemma 4.8. Let $m(B)$ denote the monad part
of $B$ and $c(B)$ the comonad part of $B$, and set $\chi(*)=X$. We find that
$\psi:m(B^{\prime})X\Rightarrow Xm(B)$ is a distributive law with respect to
monads, and that $\phi:Xc(B)\Rightarrow c(B^{\prime})X$ is a distributive law
with respect to comonads.

It is a nice exercise to prove the following lemma that we will use to pursue
with this example.

###### Lemma 5.6

Let $B$ be both a monad (to which we refer to as $m(B)$) and a comonad (we
refer to it as $c(B)$) and suppose that $\nu:BB\Rightarrow BB$ is a
distributive law, both on left and right side, with respect to monad $m(B)$
and with respect to comonad $c(B)$ (this means four distributive laws). Then

$\lambda:=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\nu$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$

is a distributive law on the left both with respect to monads and comonads,
that is:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(0.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\put(12.0,48.0){\oval(12.0,18.0)[b]} \put(12.0,39.0){\line(0,-2){3.0}}
\qbezier(36.0,48.0)(36.0,44.0)(30.0,42.0)\qbezier(30.0,42.0)(24.0,40.0)(24.0,36.0)
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(13.0,25.0){\line(1,0){12.0}} \put(13.0,35.0){\line(1,0){12.0}}
\put(13.0,35.0){\line(0,-1){10.0}}
\put(30.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda$}}
\put(30.0,36.0){\line(0,-1){1.0}} \put(30.0,24.0){\line(0,1){1.0}}
\put(35.0,25.0){\line(-1,0){12.0}} \put(35.0,35.0){\line(-1,0){12.0}}
\put(35.0,35.0){\line(0,-1){10.0}} \put(18.0,24.0){\line(0,-1){12.0}}
\put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(48.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\qbezier(12.0,48.0)(12.0,44.0)(15.0,42.0)\qbezier(15.0,42.0)(18.0,40.0)(18.0,36.0)
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\end{minipage}=\begin{minipage}{36.0pt}
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\put(6.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
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\put(18.0,30.0){\circle{5.0}} \put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle
m(B).$}} \put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (37)

Continuing with the Example, we have that similarly $\nu^{\prime}$ induces
$\lambda^{\prime}$. Moreover, the compatibilities

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\end{minipage}\qquad\text{and}\qquad\begin{minipage}{24.0pt}
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\end{minipage}=\begin{minipage}{36.0pt}
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\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (38)

hold. Then
$(X,\psi,\phi):({\mathcal{A}},m(B),c(B),\lambda)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}({\mathcal{A}}^{\prime},m(B^{\prime}),c(B^{\prime}),\lambda^{\prime})$
is a 1-cell in the 2-category of mixed distributive laws
$\operatorname{Dist}({\mathcal{K}})$ of [28, Definition 6.2]. (In the specific
case when $X$ is a left $m(B^{\prime})$-module and left
$c(B^{\prime})$-comodule, and $\psi$ and $\phi$ are given by
$\psi=\begin{minipage}{36.0pt}
\begin{picture}(36.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(6.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B^{\prime})$}}
\put(18.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle X$}}
\put(6.0,24.0){\line(0,-1){4.0}} \put(6.0,20.0){\line(1,0){12.0}}
\put(6.0,20.0){\line(3,-1){12.0}} \put(18.0,24.0){\line(0,-1){12.0}} \
\put(30.0,12.0){\line(0,1){6.0}} \put(30.0,18.0){\circle{5.0}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle X$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle m(B)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ and $\phi=\begin{minipage}{36.0pt}
\begin{picture}(36.0,36.0)\put(0.0,24.0){\line(1,0){0.0}}
\put(18.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle X$}}
\put(30.0,28.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B)$}}
\put(6.0,12.0){\line(0,1){4.0}} \put(6.0,16.0){\line(1,0){12.0}}
\put(6.0,16.0){\line(3,1){12.0}} \put(18.0,12.0){\line(0,1){12.0}}
\put(30.0,24.0){\line(0,-1){6.0}} \put(30.0,18.0){\circle{5.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle c(B^{\prime})$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle X$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$, the two expressions in (38) are equivalent and one recovers a
particular form of $\lambda$-bialgebras by Turin and Plotkin, [32, Section
7.2].)

Suppose that $B$ is a bialgebra in a braided monoidal category
${\mathcal{C}}$. A (left) Yetter-Drinfel‘d module over $B$ is an object $M$
together with a (left) action $B\otimes M\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}M$ and a (left) coaction
$M\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B\otimes M$ of $B$ subject to the
compatibility condition:

$\begin{minipage}{36.0pt}
\begin{picture}(36.0,96.0)\put(0.0,84.0){\line(1,0){0.0}}
\put(12.0,88.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(30.0,88.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(12.0,72.0){\oval(12.0,18.0)[t]} \put(12.0,84.0){\line(0,-1){3.0}}
\put(30.0,84.0){\line(0,-1){12.0}} \put(6.0,72.0){\line(0,-1){12.0}}
\qbezier(18.0,72.0)(18.0,68.0)(24.0,66.0)\qbezier(24.0,66.0)(30.0,64.0)(30.0,60.0)\qbezier(18.0,60.0)(18.0,64.0)(22.0,65.0)\qbezier(30.0,72.0)(30.0,68.0)(26.0,67.0)\put(6.0,60.0){\line(0,-1){4.0}}
\put(6.0,56.0){\line(1,0){12.0}} \put(6.0,56.0){\line(3,-1){12.0}}
\put(18.0,60.0){\line(0,-1){12.0}} \put(30.0,60.0){\line(0,-1){24.0}}
\put(6.0,36.0){\line(0,1){4.0}} \put(6.0,40.0){\line(1,0){12.0}}
\put(6.0,40.0){\line(3,1){12.0}} \put(18.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,36.0){\line(0,1){4.0}} \put(30.0,40.0){\line(1,0){12.0}}
\put(30.0,40.0){\line(3,1){12.0}} \put(42.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}}
\put(30.0,24.0){\line(0,-1){4.0}} \put(30.0,20.0){\line(1,0){12.0}}
\put(30.0,20.0){\line(3,-1){12.0}} \put(42.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M.$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}$ (39)

The category of (left) Yetter-Drinfel‘d modules over $B$ in ${\mathcal{C}}$
and left $B$-linear and $B$-colinear morphisms we denote by
${}^{B}_{B}{\mathcal{Y}D}({\mathcal{C}})$.

###### Remark 5.7

Observe that the antipode, i.e. a Hopf algebra structure on a bialgebra in the
context of Yetter-Drinfel’d modules, is used in the following two instances.
One is to construct the inverse for the braiding of the respective category.
Another one is to formulate an equivalent condition to (39). Thus, the
category of Yetter-Drinfel’d modules over a bialgebra is monoidal and even it
has a pre-braiding (non-invertible), given by:

$\scriptstyle M$$\scriptstyle N$$\scriptstyle N$$\scriptstyle M.$

###### Example 5.8

Consider two braided monoidal categories ${\mathcal{C}}$ and ${\mathcal{D}}$
and two bialgebras $B_{0},B_{1}$ in ${\mathcal{D}}$. These give rise to two
bilax (and bimonoidal) functors
$F_{B_{0}},F_{B_{1}}:Del({\mathcal{C}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}Del({\mathcal{D}})$ as in Example 4.11.
The bilax natural transformation
$\chi:{\mathcal{F}}_{0}\Rightarrow{\mathcal{F}}_{1}:{\mathcal{K}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ from Example 5.2
corresponds to a generalized notion of a Yetter-Drinfel‘d module over $B_{1}$,
which in view of the above we call strong Yetter-Drinfel‘d modules.

###### Proposition 5.9

Any Yetter-Drinfel’d module $M$ over a bialgebra $B^{\prime}$ in a braided
monoidal category $({\mathcal{C}},\Phi)$ comes from a bilax natural
transformation of Example 5.8 where $\psi$ and $\phi$ are given by

$\psi=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle B^{\prime}$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(6.0,24.0){\line(0,-1){4.0}}
\put(6.0,20.0){\line(1,0){12.0}} \put(6.0,20.0){\line(3,-1){12.0}}
\put(18.0,24.0){\line(0,-1){12.0}} \put(30.0,24.0){\line(0,-1){1.0}}
\put(30.0,12.0){\line(0,1){1.0}}
\put(25.0,13.0){\framebox(10.0,10.0)[]{$\scriptstyle\hskip 1.70709ptj^{-1}$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\hskip 4.55254pt,\quad\phi=\begin{minipage}{36.0pt}
\begin{picture}(36.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(18.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(30.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle B$}}
\put(6.0,36.0){\line(0,1){4.0}} \put(6.0,40.0){\line(1,0){12.0}}
\put(6.0,40.0){\line(3,1){12.0}} \put(18.0,36.0){\line(0,1){12.0}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(25.0,37.0){\framebox(10.0,10.0)[]{$\scriptstyle j$}}
\put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle B^{\prime}$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\hskip 4.55254pt,\quad$ (40)

for any bialgebra isomorphism $j:B\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}B^{\prime}$. (More precisely, from a
bilax endo-transformation with $j=id_{B}$ in (40).)

###### Proof.

The notation in these two diagrams is the usual one for braided monoidal
categories, concretely  and  stand for the (co)multiplication of $B^{\prime}$.
That the given $\psi$ and $\phi$ are desired distributive laws (i.e. (co)lax
natural transformations) it was proved at the beginning of [10, Section 5.1],
though for $B=B^{\prime}$ and trivial $j$. The algebra (resp. coalgebra)
morphism property of $j^{-1}$ (resp. $j$) makes (40) the desired distributive
laws for nontrivial $j$. The first claim now follows from [11, Proposition
7.5], whose conditions are fulfilled since ${\mathcal{C}}$ is braided. Set
$\nu_{F,X}=(M\otimes j^{-1})\Phi_{F\hskip 1.70709pt^{\prime},X}$ and
$\nu_{X,F}=\Phi_{X,F\hskip 1.70709pt^{\prime}}(M\otimes j)$. The second claim
follows from Corollary 7.6 of loc. cit.. ∎∎

###### Example 5.10

Let $(\chi,\psi,\phi)$ be a bilax natural transformation between bilax
functors with compatible Yang Baxter operators
$\chi:{\mathcal{F}}\Rightarrow{\mathcal{F}}^{\prime}:Del({\mathcal{C}})\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}Del({\mathcal{D}})$ where
${\mathcal{C}}$ and ${\mathcal{D}}$ are braided monoidal categories with
braidings $\Phi_{\mathcal{C}}$ and $\Phi_{\mathcal{D}}$, respectively. Then
$F:={\mathcal{F}}_{*,*}$ and $G:={\mathcal{F}}^{\prime}_{*,*}$ are bimonoidal
functors ${\mathcal{C}}\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{D}}$ as in Example 4.3,
$\chi(*)=M$ is an object in ${\mathcal{D}}$ and there are morphisms

$\psi_{X}:G(X)\otimes M\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}M\otimes
F(X)\quad\text{and}\quad\phi_{X}:M\otimes
F(X)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}G(X)\otimes M$

natural in $X\in{\mathcal{C}}$, where $\psi$ is a distributive law for the
monoidal functor structures, and $\phi$ is a distributive law for the
comonoidal functor structures, so that the left identity below holds, and
consequently the one next to it:

$\begin{minipage}{48.0pt}
\begin{picture}(48.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(-6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(XY)$}}
\put(18.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(42.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Z)$}}
\qbezier(0.0,72.0)(0.0,68.0)(3.0,66.0)\qbezier(3.0,66.0)(6.0,64.0)(6.0,60.0)
\put(18.0,72.0){\line(0,-1){12.0}}
\qbezier(42.0,72.0)(42.0,64.0)(36.0,60.0)\qbezier(36.0,60.0)(30.0,56.0)(30.0,48.0)
\put(6.0,60.0){\line(0,-1){1.0}} \put(6.0,48.0){\line(0,1){1.0}}
\put(1.0,49.0){\line(1,0){12.0}} \put(1.0,59.0){\line(1,0){12.0}}
\put(1.0,59.0){\line(0,-1){10.0}}
\put(18.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{XY}$}}
\put(18.0,60.0){\line(0,-1){1.0}} \put(18.0,48.0){\line(0,1){1.0}}
\put(23.0,49.0){\line(-1,0){12.0}} \put(23.0,59.0){\line(-1,0){12.0}}
\put(23.0,59.0){\line(0,-1){10.0}} \put(6.0,48.0){\line(0,-1){12.0}}
\put(18.0,48.0){\line(0,-1){1.0}} \put(18.0,36.0){\line(0,1){1.0}}
\put(13.0,37.0){\line(1,0){12.0}} \put(13.0,47.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda_{XY,Z}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
\put(35.0,47.0){\line(0,-1){10.0}} \put(6.0,36.0){\line(0,-1){1.0}}
\put(6.0,24.0){\line(0,1){1.0}} \put(1.0,25.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(1,0){12.0}} \put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{XZ}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
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\put(18.0,24.0){\line(0,-1){12.0}}
\put(-6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(XZ)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Y)$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{72.0pt}
\begin{picture}(72.0,84.0)\put(0.0,72.0){\line(1,0){0.0}}
\put(6.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(XY)$}}
\put(30.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(54.0,76.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Z)$}}
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\put(30.0,72.0){\line(0,-1){12.0}}
\qbezier(54.0,72.0)(54.0,68.0)(48.0,66.0)\qbezier(48.0,66.0)(42.0,64.0)(42.0,60.0)
\put(30.0,60.0){\line(0,-1){1.0}} \put(30.0,48.0){\line(0,1){1.0}}
\put(25.0,49.0){\line(1,0){12.0}} \put(25.0,59.0){\line(1,0){12.0}}
\put(25.0,59.0){\line(0,-1){10.0}}
\put(42.0,54.0){\makebox(0.0,0.0)[]{$\scriptstyle\phi_{Z}$}}
\put(42.0,60.0){\line(0,-1){1.0}} \put(42.0,48.0){\line(0,1){1.0}}
\put(47.0,49.0){\line(-1,0){12.0}} \put(47.0,59.0){\line(-1,0){12.0}}
\put(47.0,59.0){\line(0,-1){10.0}} \put(18.0,48.0){\line(0,-1){1.0}}
\put(18.0,36.0){\line(0,1){1.0}} \put(13.0,37.0){\line(1,0){12.0}}
\put(13.0,47.0){\line(1,0){12.0}} \put(13.0,47.0){\line(0,-1){10.0}}
\put(30.0,42.0){\makebox(0.0,0.0)[]{$\scriptstyle\lambda^{\prime}_{XY,Z}$}}
\put(30.0,48.0){\line(0,-1){1.0}} \put(30.0,36.0){\line(0,1){1.0}}
\put(35.0,37.0){\line(-1,0){12.0}} \put(35.0,47.0){\line(-1,0){12.0}}
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\put(25.0,35.0){\line(0,-1){10.0}}
\put(42.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{Y}$}}
\put(42.0,36.0){\line(0,-1){1.0}} \put(42.0,24.0){\line(0,1){1.0}}
\put(47.0,25.0){\line(-1,0){12.0}} \put(47.0,35.0){\line(-1,0){12.0}}
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\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(XZ)$}}
\put(30.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(54.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Y),$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\qquad\begin{minipage}{24.0pt}
\begin{picture}(24.0,48.0)\put(0.0,36.0){\line(1,0){0.0}}
\put(6.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(18.0,40.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(6.0,36.0){\line(0,-1){1.0}} \put(6.0,24.0){\line(0,1){1.0}}
\put(1.0,25.0){\line(1,0){12.0}} \put(1.0,35.0){\line(1,0){12.0}}
\put(1.0,35.0){\line(0,-1){10.0}}
\put(18.0,30.0){\makebox(0.0,0.0)[]{$\scriptstyle\psi_{X}$}}
\put(18.0,36.0){\line(0,-1){1.0}} \put(18.0,24.0){\line(0,1){1.0}}
\put(23.0,25.0){\line(-1,0){12.0}} \put(23.0,35.0){\line(-1,0){12.0}}
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\put(23.0,23.0){\line(0,-1){10.0}}
\put(6.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(18.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(42.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle M$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,36.0){\line(0,1){4.0}} \put(30.0,40.0){\line(1,0){12.0}}
\put(30.0,40.0){\line(3,1){12.0}} \put(42.0,36.0){\line(0,1){12.0}}
\put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(42.0,36.0){\line(0,-1){12.0}}
\put(12.0,24.0){\oval(12.0,18.0)[b]} \put(12.0,15.0){\line(0,-2){3.0}}
\put(30.0,24.0){\line(0,-1){4.0}} \put(30.0,20.0){\line(1,0){12.0}}
\put(30.0,20.0){\line(3,-1){12.0}} \put(42.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle G(X)$}}
\put(42.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle M,$}}
\put(0.0,12.0){\line(1,0){0.0}} \end{picture} {\vskip 9.68745pt}
\end{minipage}\quad\text{with}\quad\lambda_{XY,Z}:=\begin{minipage}{48.0pt}
\begin{picture}(48.0,60.0)\put(0.0,48.0){\line(1,0){0.0}}
\put(12.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(XY)$}}
\put(36.0,52.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Z)$}}
\put(12.0,36.0){\oval(12.0,18.0)[t]} \put(12.0,48.0){\line(0,-1){3.0}}
\put(30.0,48.0){\line(0,-1){12.0}} \put(6.0,36.0){\line(0,-1){12.0}}
\qbezier(18.0,36.0)(18.0,32.0)(24.0,30.0)\qbezier(24.0,30.0)(30.0,28.0)(30.0,24.0)\qbezier(18.0,24.0)(18.0,28.0)(22.0,29.0)\qbezier(30.0,36.0)(30.0,32.0)(26.0,31.0)\put(12.0,24.0){\oval(12.0,18.0)[b]}
\put(12.0,15.0){\line(0,-2){3.0}} \put(30.0,24.0){\line(0,-1){12.0}}
\put(12.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(XZ)$}}
\put(36.0,0.0){\makebox(0.0,0.0)[b]{$\scriptstyle F(Y).$}}
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\end{minipage}$

For bialgebras $B$ in ${\mathcal{C}}$ by Proposition 4.9 $F(B),G(B)$ are
bialgebras in ${\mathcal{D}}$, and if $\psi_{B},\phi_{B}$ are of the form as
in (40), we recover classical Yetter-Drinfel‘d modules in ${\mathcal{C}}$.

The bilax natural transformations, i.e. identities (34) and (35), offer the
following point of view. Given any monoidal category ${\mathcal{D}}$, when one
considers the center category ${\mathcal{Z}}_{l}^{w}({\mathcal{D}})$, one is
given a family of colax transformations $\phi$. In particular, when
${\mathcal{D}}={}_{H}{\mathcal{C}}$ the category of modules over a bialgebra
or a Hopf algebra $H$ in a braided monoidal category ${\mathcal{C}}$, one is
able to construct lax transformations $\psi$ (as in (40)) so that the given
$\phi$ and this $\psi$ obey (35) - since $\phi$ is $H$-linear, being a
morphism in ${}_{H}{\mathcal{C}}$. Similarly, considering
${\mathcal{Z}}_{r}^{w}({}^{H}{\mathcal{C}})$ one is given $\psi$’s and one
constructs $\phi$’s, so that they together obey (35). As in the proof of the
above Proposition (that is, as proved in [11, Proposition 7.5]), in this
setting the bilax condition (34) follows.

###  5.2 Bilax modifications

We finally introduce:

###### Definition 5.11

Let
$\chi,\chi^{\prime}:({\mathcal{F}},\nu)\Rightarrow({\mathcal{F}}^{\prime},\nu^{\prime}):({\mathcal{K}},c)\allowbreak\mathrel{\mathop{\hbox
to10.79999pt{\rightarrowfill}}\limits}{\mathcal{K}}^{\prime}$ be bilax natural
transformations. A bilax modification $a:\chi\Rrightarrow\chi^{\prime}$ is a
collection of 2-cells $(a(A))_{A\in{\mathcal{O}b}({\mathcal{K}})}$ satisfying
equations: