2.1 Some computational tools for globular 2-cells

For 2-cells of the form of a as below we will say that they are horizontally globular, while for those of the form of b we will say that they are vertically globular.

For computations that involve only horizontally or only vertically globular 2-cells, one may use the string diagrams for 2-categories, which are simpler to write and easier to handle in the proofs. This way the string diagrams are to be understood as living in the underlying horizontal (or vertical) 2-category of the double category. In this article our string diagrams are read from top to bottom and from left to right. We present briefly the notation that will be used in this work.

Let $\mathcal{F}$ be a lax functor and $\mathcal{G}$ an oplax functor between 2-categories. We depict their lax, respectively, oplax structures by string diagrams in the following way:

where f,g are composable 1-cells and A a 0-cell in the domain 2-category.

Recall that for a 1-cell $f \colon A\to B$ in a 2-category its left adjoint is a 1-cell $u \colon B \to A$ together with two 2-cells $\eta: \operatorname {id}_A\to u f$ and $\varepsilon: f u \to \operatorname {id}_B$ such that

\begin{equation*} \frac{[\eta\vert\operatorname {Id}_f]}{[\operatorname {Id}_f\vert\varepsilon]} = \operatorname {id}_f \qquad \qquad \text{and} \qquad \qquad \frac{[\operatorname {Id}_u\vert\eta]}{[\varepsilon\vert\operatorname {Id}_u]} = \operatorname {id}_u.\end{equation*}

In string diagrams, we will write and they satisfy the laws:

It is a folklore that any equivalence between 1-cells in a 2-category can be made into an adjoint equivalence.

We record now two identities regarding mates that will be used in Proposition 8.5. Suppose there are equivalence 1-cells f,f’ in a 2-category with their respective adjoint quasi-inverses $f^\bullet$ and $(f')^\bullet$ . Let further 1-cells g,g’ be given so that the compositions g’f and f’g are possible and suppose that there is an invertible 2-cell $\alpha: f'g\Rightarrow g'f$ (with inverse $\alpha^{-1}: g'f\Rightarrow f'g$ ). The mates $\alpha^\bullet$ and $(\alpha^{-1})^\bullet$ of $\alpha$ and its inverse are given by

where left-most stands for $\eta$ for f’ and the other one stands for $\varepsilon^{-1}$ for f, and the first appearing stands for $\varepsilon$ for f and the other one for $\eta^{-1}$ for f’. Then given $\alpha_1=\alpha:f'g\Rightarrow g'f$ and $\alpha_2:fh\Rightarrow h'f'$ (with $\alpha_2^\bullet: h(f')^\bullet\Rightarrow f^\bullet h'$ ) one has identities

and one also has

where in the left-hand side identity denotes $\eta$ for f and denotes $\eta^{-1}$ for f’, and in the right-hand side identity denotes $\varepsilon^{-1}$ for f and denotes $\varepsilon$ for f’. (The 2-cell $\alpha$ here will be used in Proposition 8.5 as a 2-cell component of a horizontal pseudonatural transformation $\alpha_{A,X,-}^3$ , the 1-cells f,f’ as 1h-cells $\alpha_{A,X,B}^3$ and $\alpha_{A,X,B'}^3$ , and the 1-cells g,g’ as 1h-cells $AB\ltimes h$ and $A(X\ltimes h)$ , respectively, and similarly.)

2.2 The double category of double functors

We recall here the definitions of lax/pseudo double functors, horizontal and vertical (op)lax/pseudo transformations and modifications.

1. • (functoriality in vertical morphisms)

   $$ \boldsymbol{(lx.f.v1)} \quad \qquad \frac{F(u)}{F(u')}=F(\frac{u}{u'}), \qquad \qquad \boldsymbol{(lx.f.v2)} \quad \qquad F(1^A)=1^{F(A)};$$

2. • (functoriality in squares)

   $$ \boldsymbol{(lx.f.s1)} \quad \qquad F(\frac{\omega}{\zeta})=\frac{F(\omega)}{F(\zeta)}, \qquad \qquad \boldsymbol{(lx.f.s2)} \quad \qquad F(Id_f)=Id_{F(f)};$$

3. • (coherence with compositors and unitors)

   $$ \boldsymbol{(lx.f.cmp)} \hspace{2cm}\qquad \frac{[F_{gf}\vert Id_{F(h)}]}{F_{h,gf}}=\frac{[Id_{F(f)}\vert F_{hg}]}{F_{hg,f}} $$
   $$ \boldsymbol{(lx.f.u)} \hspace{2cm}\qquad \frac{[F_A\vert Id_{F(f)}]}{F_{f1_A}}=\operatorname {Id}_{F(f)}=\frac{[Id_{F(f)}\vert F_B]}{F_{1_Bf}};$$

4. • (naturality of the compositor)

• (naturality of the unitor)

where u,u’ are composable 1v-cells, $\omega, \zeta$ vertically composable 2-cells, $\alpha, \beta$ horizontally composable 2-cells, and f,g,h composable 1h-cells.

A pseudodouble functor is a lax double functor whose compositor and unitor 2-cells are invertible.

We now define horizontal oplax and vertical lax transformations between lax double functors and their modifications. The reason why we introduce these kinds of transformations is explained at the beginning of Subsection 6.1.

1. (1) for every 0-cell A in $\mathbb{A}$ a 1h-cell $\alpha(A)\colon F(A)\to G(A)$ in $\mathbb{B}$ ,

2. (2) for every 1v-cell $u\colon A\to A'$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

3. (3) for every 1h-cell $f\colon A\to B$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   
   so that the following are satisfied:

4. • (coherence with compositors for $\delta_{\alpha,-}$ ): for any composable 1h-cells f and g in $\mathbb{A}$ the 2-cell $\delta_{\alpha,gf}$ satisfies:

(h.o.t.-1)

(coherence with unitors for $\delta_{\alpha,-}$ ): for any object $A\in\mathbb{A}$ :

(h.o.t.-2)

1. • (coherence with vertical composition and identity for $\alpha^\bullet$ ): for any composable 1v-cells u and v in $\mathbb{A}$ :

   $$ \boldsymbol{(h.o.t.-3)} \qquad \alpha^{\frac{u}{v}}=\frac{\alpha^u}{\alpha^v}\quad \qquad \text{ and}\quad \qquad \boldsymbol{(h.o.t.-4)} \qquad \alpha^{1^A}=\operatorname {Id}_{\alpha(A)};$$

2. • (oplax naturality of 2-cells): for every 2-cell in $\mathbb{A}$ the following identity in $\mathbb{B}$ must hold:

$\boldsymbol{(h.o.t.-5)}$

A horizontal lax transformation is defined by using the opposite direction of the 2-cells $\delta_{\alpha,f}$ in item 3 and accommodating the axioms correspondingly.

A horizontal pseudonatural transformation is a horizontal oplax transformation for which the 2-cells $\delta_{\alpha,f}$ are isomorphisms. A horizontal strict transformation is a horizontal oplax transformation for which the 2-cells $\delta_{\alpha,f}$ are identities.

The axioms of a horizontal lax transformation we denote by (h.l.t.-1)-(h.l.t.-5). Indeed, note that from the five axioms only the three axioms (h.o.t.-1), (h.o.t.-2) and (h.o.t.-5) are changed into (h.l.t.-1), (h.l.t.-2) and (h.l.t.-5).

1. (1) for every 0-cell A in $\mathbb{A}$ a 1h-cell in $\mathbb{B}$ :

   $$\bigg(\frac{\alpha}{\beta}\bigg)(A)=\big( F(A)\stackrel{\alpha(A)}{\longrightarrow}G(A) \stackrel{\beta(A)}{\longrightarrow} H(A) \big),$$

2. (2) for every 1v-cell $u\colon A\to A'$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

3. (3) for every 1h-cell $f\colon A\to B$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

1. (1) a 1v-cell $\alpha_0(A)\colon F(A)\to G(A)$ in $\mathbb{B}$ for every 0-cell A in $\mathbb{A}$ ;

2. (2) for every 1h-cell $f\colon A\to B$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

3. (3) for every 1v-cell $u\colon A\to A'$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

which need to satisfy:

1. • (coherence with compositors for $(\alpha_0)_\bullet$ ): for any composable 1h-cells f and g in $\mathbb{A}$ :

(coherence with unitors for $(\alpha_0)_\bullet$ ): for any object A in $\mathbb{A}$ :

1. • (coherence with vertical composition for $\alpha_0^\bullet$ ): for any composable 1v-cells u and v in $\mathbb{A}$ :

(coherence with vertical identity for $\alpha^\bullet$ ): for any object A in $\mathbb{A}$ :

$$ \boldsymbol{(v.l.t.\mbox{-} 4)} \qquad \qquad \alpha_0^{1^A}=\operatorname {Id}^{\alpha_0(A)} \hspace{4cm}$$

1. • (lax naturality of 2-cells): for every 2-cell in $\mathbb{A}$ the following identity in $\mathbb{B}$ must hold:

   

A vertical oplax transformation is defined by using the opposite direction of the 2-cells $\alpha_0^u$ in item 3 and accommodating the axioms correspondingly.

A vertical lax transformation is called a vertical pseudonatural transformation if the globular 2-cells $\alpha_0^u$ are horizontally invertible for all 1v-cells u in $\mathbb{A}$ .

It is called a vertical strict transformation if all the globular 2-cells $\alpha_0^u$ are identities. A vertical strict transformation is said to be invertible if the 2-cells $(\alpha_0)_f$ for all 1h-cells f in $\mathbb{A}$ are vertically invertible (this includes the condition that the 1v-cells $\alpha_0(A)$ are invertible for all $A\in\mathbb{A}$ ).

The axioms for a vertical oplax transformation we denote by (v.o.t.- 1) - (v.o.t.- 5). They are made explicit in Femi´c (2023, Definition 2.5). As in the horizontal case, only the axioms (v.l.t.- 1), (v.l.t.- 3) and (v.l.t.- 5) are changed into (v.o.t.- 1), (v.o.t.- 3) and (v.o.t.- 5).

Similarly as in Femić (Reference Femić2023, Lemma 2.6) and Femić (Reference Femić2024, Lemma 3.8) one has:

for every 0-cell A in $\mathbb{A}$ a 1v-cell on the left below, and for every 1h-cell $f\colon A\to B$ in $\mathbb{A}$ a 2-cell on the right below, both in $\mathbb{B}$ :

We will need modifications between horizontal and vertical pseudonatural transformations. Their definition is formally the same as of modifications of more general transformations that we cite next from Femić (Reference Femić2023, Definition 2.7).

which satisfy the following rules:

(m.ho-vl.-1) for every 1h-cell f, we have

and

(m.ho-vl.-2) for every 1v-cell u, we have

Similarly, one defines a modification between two horizontal lax transformations and two vertical oplax transformations, whose axioms we denote by (m.hl-vo.-1) and (m.hl-vo.-2). We will use the axioms (h.l.t.-1) - (h.l.t.5), (v.o.t.- 1) - (v.o.t.- 5) and (m.hl-vo.-1) - (m.hl-vo.-2) in the proof of Theorem 6.6 and in Table 7.

Taking, so to say, a horizontal and a vertical restriction of modifications in Definition 2.7, we obtain the definitions of:

1. • horizontal modifications or modifications between horizontal (oplax) transformations given by families of (horizontally globular) 2-cells

(2)

and axioms (m.ho.-1) and (m.ho.-2) obtained from (m.ho-vl.-1) and (m.ho-vl.-2) by ignoring the 2-cells $(\alpha_0)_f, (\beta_0)_f, \alpha_0^u$ and $\beta_0^u$ , and

1. • vertical modifications or modifications between vertical (lax) transformations given by families of (vertically globular) 2-cells

(3)

and axioms (m.vl.-1) and (m.vl.-2) obtained from (m.ho-vl.-1) and (m.ho-vl.-2) by ignoring the 2-cells $\delta_{\alpha,f}, \delta_{\beta,f}, \alpha^u$ and $\beta^u$ .

We will use the following fact that is straightforwardly proved (see also Femić (Reference Femić2024, Lemma 3.4)).

1. a) The 2-cells $\delta_{H(\omega),f}=\begin{array}{@{\,}c@{\,}}H_{lx}\\ \hline H(\delta_{\omega,f})\\ \hline H_{oplx} \end{array}$ and $H(\omega)^u=H(\omega^u)$ , where $H_{lx}$ and $H_{oplx}$ represent the lax and oplax functor structure of H applied to the corresponding 1h-cells, respectively, define a horizontal pseudonatural transformation $H(\omega):HF\Rightarrow HG$ of lax/pseudo double functors.

2. b) The 2-cells $\delta_{(\omega L),f}=\delta_{\omega,L(f)}$ and $(\omega L)^u=\omega^{L(u)}$ define a horizontal pseudonatural transformation $\omega L:FL\Rightarrow GL$ of lax/pseudo double functors.

An analogous result holds for vertical pseudonatural transformations. Its proof is even simpler, as in our convention the vertical direction (for lax/pseudo double functors) is strict.

1. a) The 2-cells $H(\alpha)_f=H(\alpha_f)$ and $H(\omega)^u=H(\omega^u)$ define a vertical pseudonatural transformation $H(\alpha):HF\Rightarrow HG$ of lax/pseudo double functors.

2. b) The 2-cells $\delta_{\alpha L,f}=\delta_{\alpha, L(f)}$ and $(\alpha L)^u=\alpha^{L(u)}$ define a vertical pseudonatural transformation $\alpha L:FL\Rightarrow GL$ of lax/pseudo double functors.

2.3 Companions and conjoints

Recall that a companion for a 1v-cell $u:A\to B$ is a 1h-cell $\hat{u}:A\to B$ together with two 2-cells $\varepsilon$ and $\eta$ as below satisfying $[\eta\vert\varepsilon]=\operatorname {Id}_{\hat{u}}$ and $\frac{\eta}{\varepsilon}=\operatorname {Id}_{u}$ . On the other hand, a conjoint for a 1v-cell $u: A\to B$ is a 1h-cell $\check{u}:B\to A$ together with two 2-cells $\varepsilon^*$ and $\eta^*$ as below satisfying $[\varepsilon^*\vert\eta^*]=\operatorname {Id}_{\check{u}}$ and $\frac{\eta^*}{\varepsilon^*}=\operatorname {Id}^{u}$ .

Companions are unique up to a unique globular isomorphism. This and more properties of companions and conjoints we will recall in Subsection 2.4, see also Grandis and Paré (Reference Grandis and Paré2004, Section 1.2), Shulman (Reference Shulman2010, Section 3). For the moment, we focus on the following few facts about them.

The unit $\tilde\eta$ and counit $\tilde\varepsilon$ of the adjunction are given via $\tilde\eta=[\eta\vert\eta^*]$ and $\tilde\varepsilon=[\varepsilon^*\vert\varepsilon]$ and their inverses by $\tilde\eta^{-1}=\frac{[\varepsilon\vert\varepsilon^*]}{[\operatorname {Id}^{u^{-1}}\vert\operatorname {Id}^{u^{-1}}]}$ and $\tilde\varepsilon^{-1}=[\eta^*\vert\eta]$ .

We record that also the following identities hold true

(4) \begin{equation} \frac{\eta_u}{\varepsilon^*_{u^{-1}}}=1, \quad \frac{\varepsilon_u}{\eta^*_{u^{-1}}}=\operatorname {Id}_{\hat{u}},\quad \frac{\varepsilon^*_{u^{-1}}}{\eta}=\operatorname {Id}_{\hat{u}}, \quad \frac{\eta^*_{u^{-1}}}{\varepsilon_u}=1.\end{equation}

We will need existence of companions and conjoints only for 1v-components of invertible vertical strict transformations. In this case by Corollary 2.12, it is sufficient to require the existence of companions. The following result will be very important for us.

1. (1) The following data define a horizontal pseudonatural transformation $\alpha_1:F\Rightarrow G$ :

   1. a) for all 1v-cell components $\alpha_0(A)$ of $\alpha_0$ a fixed choice of companions (and conjoints) in $\mathbb{E}$ , (we denote their companions by $\alpha_1(A)$ for every 0-cell A of $\mathbb{D}$ , the corresponding 2-cells by $\varepsilon^\alpha_A$ and $\eta^\alpha_A$ , and also by $\varepsilon^*_A$ and $\eta^*_A$ the 2-cells related to conjoints of the inverse of $\alpha_0(A)$ );

   2. b) the 2-cells

      

      in $\mathbb{E}$ for every 1h-cell $f:A\to B$ in $\mathbb{D}$ ;

   3. c) the 2-cells

      

      in $\mathbb{E}$ for every 1v-cell $u:A\to \tilde A$ in $\mathbb{D}$ ;

   4. d) the inverses of the 2-cells $\delta_{\alpha_1,f}$ are given by

      

2. (2) The 1h-cells $\alpha_1(A)$ are adjoint equivalence 1-cells in $\mathcal{H}(\mathbb{E})$ .

3. (3) The data from 1. define a horizontal equivalence $\alpha_1:F\Rightarrow G:\mathbb{D}\to \mathbb{E}$ and a pseudonatural equivalence $\mathcal{H}(\alpha_1):\mathcal{H}(F)\Rightarrow \mathcal{H}(G):\mathcal{H}(\mathbb{D})\to \mathcal{H}(\mathbb{E})$ .

The above proposition in particular holds for invertible vertical strict transformations (recall Definition 2.5) whose 1v-cell components have companions. As such transformations will be crucial in this work, we introduce some terminology.

Those vertical transformations all of whose 1v-cell components are liftable we will call companion-liftable (or shortly liftable) vertical transformations.

A horizontal pseudonatural equivalence $\alpha_1$ obtained in Proposition 2.14 from an invertible liftable vertical strict transformation $\alpha_0$ we will call a companion-lift of $\alpha_0$ .

In Shulman (Reference Shulman2010, Theorem 4.6), it is shown that the above-described assignment of bicategorical pseudonatural equivalences to invertible liftable vertical strict transformations, i.e., a companion-lifting of such vertical transformations, is functorial. Lemma 2.13 and the above proposition, together with a reasoning that we will highlight in Proposition 2.22 further below, were crucial pieces on which relies the proof of Shulman’s Shulman (Reference Shulman2010, Theorem 5.1). Let us recall what it means for a double category to be monoidal due to Shulman, in our terminology.

Definition 2.16 Shulman (2010, Definition 2.9). A monoidal double category is a double category $\mathbb{D}$ equipped with pseudodouble functors $\otimes:\mathbb{D}\times\mathbb{D}\to \mathbb{D}$ and $I:*\to \mathbb{D}$ and invertible vertical strict transformations

\begin{align*}\begin{array}{c}\alpha: \otimes\circ(\!\operatorname {Id}\times\otimes) \stackrel{\cong}{\to } \otimes\circ(\!\otimes\times\operatorname {Id}\!) \\\lambda: \otimes\circ(I\times\operatorname {Id}\!) \stackrel{\cong}{\to } \operatorname {Id} \\\rho: \otimes\circ(\!\operatorname {Id}\times I) \stackrel{\cong}{\to } \operatorname {Id}\end{array} \end{align*}

satisfying the pentagonal and three triangular axioms (via identity vertical modifications).

In any 2-category with finite products, there is a notion of a pseudomonoid. Thus, a monoidal double category in other words is a pseudomonoid in the 2-category of double categories, pseudofunctors, and vertical strict transformations.

We recall now Shulman (Reference Shulman2010, Theorem 5.1). “Isofibrancy” there supposes existence of companions for all 1v-cells in $\mathbb{D}$ , whereas one indeed needs only existence of 1v-cell components of the monoidal structure $(\alpha,\lambda,\rho)$ of $\mathbb{D}$ . We give an accordingly adapted formulation.

The proof of this theorem used a remarkable tool that we show in details in the next subsection. It will also show very important for our work, in particular in Section 10.

2.4 Lifting of modifications

In Proposition 2.14, we saw how, assuming the existence of companions, a (invertible) vertical strict transformation induces a (pseudonatural) horizontal transformation. For this, we can say that vertical (strict) transformations lift to horizontal ones. In this subsection, we will prove that a similar occurrence happens for modifications and axioms that they obey.

We start by recalling some more properties of companions.

Lemma 2.18 Shulman (2010, Lemma 3.8). Between two companions $\hat{u}:A\to B$ and $\hat{u}':A\to B$ of a 1v-cell $u:A\to B$ in $\mathbb{D}$ there is a unique globular isomorphism $\theta$ such that

(5)

It is given by

(6)

of the induced horizontal transformation $\hat{\alpha}$ is equal to $\theta_{[F(\hat{u})\vert\hat{\alpha}(B)], [\hat{\alpha}(A)\vert G(\hat{u})]}$ , and in particular, it is an isomorphism.

Let us recall the basic algebra of companions from Shulman (Reference Shulman2010, Section 3).

1. (1) An identity 1v-cell has the identity 1h-cell as a companion.

2. (2) If 1v-cells $u:A\to B$ and $v:B\to C$ have companions $\hat{u}$ and $\hat{v}$ , then $\frac{u}{v}$ has a companion $[\hat{u}\vert\hat{v}]$ .

3. (3) If u has three companions $\hat{u}, \hat{u}', \hat{u}''$ , then $\theta_{\hat{u},\hat{u}''}=\frac{\theta_{\hat{u},\hat{u}'}}{\theta_{\hat{u}',\hat{u}''}}$ .

4. (4) If 1v-cells $u:A\to B$ and $v:B\to C$ have companions $\hat{u}, \hat{u}'$ and $\hat{v},\hat{v}'$ , then $[\theta_{\hat{u},\hat{u}'} \vert\theta_{\hat{v},\hat{v}'}]=\theta_{[\hat{u},\hat{v}],[\hat{u}',\hat{v}']}$ .

5. (5) If u has a companion $\hat{u}$ , then $\theta_{\hat{u}, [\operatorname {id}\vert\hat{u}]}$ and $\theta_{\hat{u}, [\hat{u}\vert\operatorname {id}]}$ are equal to the unit constraints $\hat{u}\cong [\operatorname {id}\vert\hat{u}]$ and $\hat{u}\cong [\hat{u}\vert\operatorname {id}]$ .

6. (6) Let $F:\mathbb{D}\to \mathbb{E}$ be a pseudodouble functor between double categories and assume that u has companions $\hat{u}, \hat{u}'$ in $\mathbb{D}$ . Then:

   1. a) F(u) has companions $F(\hat{u})$ and $F(\hat{u}')$ , and

   2. b) $\theta_{F(\hat{u}),F(\hat{u}')}=F(\theta_{\hat{u},\hat{u}'})$ in $\mathbb{E}$ .

7. (7) Assume that $\mathbb{D}$ is a monoidal double category and that $u:A\to \tilde A$ and $v:B\to \tilde B$ have companions $\hat{u},\hat{u}'$ and $\hat{v},\hat{v}'$ . Then:

   1. a) $u\otimes v$ has companions $\hat{u}\otimes\hat{v}$ and $\hat{u}'\otimes\hat{v}'$ , and

   2. b) $\theta_{\hat{u},\hat{u}'} \otimes \theta_{\hat{v},\hat{v}'} = \theta_{\hat{u}\otimes\hat{v},\hat{u}'\otimes\hat{v}'}$ .

We extract the essence of the mechanism used by Shulman, which underlies the proof of his remarkable Shulman (Reference Shulman2010, Theorem 5.1) in the following proposition. Its proof is a reformulation of Shulman’s one.

Proposition 2.22. Let $\omega$ be the identity vertical modification between two vertical composites of vertical strict transformations

$$\omega: \,\, \begin{array}{@{\,}c@{\,}}\alpha_1\\ \hline ...\\ \hline \alpha_k \end{array} \,\,\, \Rrightarrow \,\,\, \begin{array}{@{\,}c@{\,}}\beta_1\\ \hline ...\\ \hline \beta_l \end{array} \,\,$$

which act between lax double functors $F\Rightarrow G:\mathbb{B}\to \mathbb{D}$ between double categories, so that all 1v-cell components $\alpha_1(A),...\alpha_k(A)$ and $\beta_1(A), ..., \beta_l(A)$ for 0-cells A in $\mathbb{B}$ , have companions in $\mathbb{D}$ . Then:

1. (1) $\omega$ induces an invertible horizontal modification $\hat{\omega}$ between the two (vertical compositions of the) induced horizontal natural transformations;

2. (2) the assignment between 2-cell components of $\omega$ and $\hat{\omega}$ is invertible;

3. (3) if $\omega_1,...,\omega_m$ are vertical modifications with the above characteristics, then any sensible equation formed by their 2-cell components $\hat{\omega}_1(A),...,\hat{\omega}_m(A)$ , for any 0-cell A in $\mathbb{B}$ , holds true.

Let A be fixed. Observe that $\omega(A)$ presents the identity between its domain and codomain composite 1v-cells for every A. We will abuse notation by writing $\omega$ both for the modification and its component 2-cells $\omega(A)$ .

Let $\omega:u\Rightarrow v$ denote $\omega$ as a vertically globular 2-cell between its composite 1v-cell components u and v. Then, u and v are 1v-cell components of the composite vertical strict transformations. The (clearly invertible) assignment $\omega\mapsto\hat{\omega}$ is given by

(7)

where $\eta_u$ is given by diagonally composing $\eta$ ’s for every component 1v-cell making the composite u, and the same for $\varepsilon_v$ . We used here part 2. of Proposition 2.21. (If some component 1v-cell in u or v is of the form $x\otimes y$ for some 1v-cells x,y, then by part 7a) of Proposition 2.21 we have $\widehat{x\otimes y}=\hat{x}\otimes\hat{y}$ .)

Since $u=v$ via $\omega$ by assumption, then $\hat{\omega}: \hat{u}\Rightarrow\hat{v}$ satisfies (5), it is a $\theta$ 2-cell and an isomorphism by Lemma 2.18 with inverse given by

On the other hand, by Lemma 2.19, it induces an invertible horizontal modification between the composite horizontal natural transformations.

If we are given any equation relating horizontally globular 2-cells $\hat{\omega}_1,...,\hat{\omega}_m$ induced from vertically globular 2-cells $\omega_1,...,\omega_m$ with the above characteristics, we have as above that every $\hat{\omega}_i, i=1,2..,m$ is given by a $\theta$ 2-cell. By properties 3. and 4. of Proposition 2.21 we have that (combinations of horizontal and vertical) composites of 2-cells $\hat{\omega}_1,...,\hat{\omega}_m$ making the two sides of the equation are both isomorphism 2-cells of the sort of $\theta$ between their common domain and codomain. The uniqueness of $\theta$ implies that the equation in question holds.

We will also need the following variation of the above claim. In it we treat $\omega$ as component 2-cells of the modification in question.

1. (1) $\omega$ induces a 2-cell $\omega^*$ defining a modification in the sense of Definition 2.7;

2. (2) there is a one-to-one correspondence between 2-cells $\omega^*$ and $\hat{\omega}$ from Proposition 2.22, so that $\omega^*$ induces a modification if and only if so does $\hat{\omega}$ ;

3. (3) if $\omega^*_1,...,\omega^*_m$ are 2-cells induced by vertically globular 2-cells $\omega_1,...,\omega_m$ with the characteristics as $\omega$ of this proposition, then any sensible equation formed by the 2-cells $\omega^*_1,...,\omega^*_m$ holds true.

(8)

(the assumption that at least two 1v-cells in a same edge of $\omega$ are nontrivial is there only to assure that we get a nonglobular 2-cell $\omega^*$ ). The proofs of all the three claims are similar and straightforward. We will comment on the proof of the third claim. The invertible assignment in part 2. is given by:

$$\omega^*\mapsto [\eta\vert\omega^*\vert\varepsilon]\qquad \text{and}\qquad \hat{\omega}\mapsto \begin{array}{@{\,}c@{\,}}[\operatorname {Id}\vert\eta]\\ \hline \hat{\omega}\\ \hline [\varepsilon\vert\operatorname {Id}] \end{array}.$$

For the third part, write any sensible equation $E^*$ formed by $\omega^*_1,...,\omega^*_m$ in terms of $\hat{\omega}_1,...,\hat{\omega}_m$ using part 2. to obtain an equation $\tilde E$ . Then $E^*$ holds true if and only if $\tilde E$ does. Observe that both vertical and horizontal composition of some $\omega^*_i$ and $\omega^*_j$ obtains the form $\begin{array}{@{\,}c@{\,}c@{\,}} [\operatorname {Id}\vert\operatorname {Id}\vert\eta]\\ \hline [\operatorname {Id}\vert\hat{\omega}_j]\\ \hline [\hat{\omega}_i\vert\operatorname {Id}]\\ \hline [\varepsilon\vert\operatorname {Id}\vert\operatorname {Id}]\end{array}$ in $\tilde E$ (or its symmetric diagonal version). Consequently, in the equation $\tilde E$ “middle $\eta$ ’s and $\varepsilon$ ’s” will cancel out and $\tilde E$ gets the form $\begin{array}{@{\,}c@{\,}}[\operatorname {Id}\vert\eta]\\ \hline \hat{E}\\ \hline [\varepsilon\vert\operatorname {Id}] \end{array}$ , i.e., $[\operatorname {Id}\vert\eta]$ and $[\varepsilon\vert\operatorname {Id}]$ are composed to both sides of $\hat{E}$ , where $\hat{E}$ is an equation as in the third claim of Proposition 2.22. We know that $\hat{E}$ holds true, hence we have the proof.

For liftable u,u’ we label the invertible assignment

(9)

for future reference.

Under assumption on existence of companions, invertibility of vertically globular 2-cells does not imply in general invertibility of squares obtained by the above described assignment $\omega\mapsto\omega^*$ . However, we have

We state for the record that although invertibility of squares implies invertibility of horizontally globular 2-cells (similar to the assignment $\omega^*\mapsto\hat{\omega}$ from Proposition 2.23), the converse holds under assumptions similar as in the above lemma.

3.1 Binoidal double categories and central cells

We generalize the definitions of a binoidal bicategory and central 1- and 2-cells in it from Paquet and Saville (Reference Paquet and Saville2023a) to double categories. We will differentiate left and right central 1-cells.

• A left central 1h-cell in $\mathbb{B}$ is a 1h-cell $f:A\to A'$ in $\mathbb{B}$ equipped with a horizontal pseudonatural transformation $f\ltimes-:A\ltimes -\to A'\ltimes-$ such that $f\ltimes B=f\rtimes B$ for all $B\in\mathbb{B}$ . Likewise, a right central 1h-cell in $\mathbb{B}$ is a 1h-cell $f:A\to A'$ in $\mathbb{B}$ equipped with a horizontal pseudonatural transformation $-\rtimes f:-\ltimes A\to -\rtimes A'$ such that $B\rtimes f=B\ltimes f$ for all $B\in\mathbb{B}$ .

A 1h-cell is said to be central if it is both left and right central.

• A left central 1v-cell in $\mathbb{B}$ is a 1v-cell $v:A\to \tilde A$ in $\mathbb{B}$ equipped with a vertical pseudonatural transformation $v\ltimes-:A\ltimes -\to \tilde A\ltimes-$ such that $v\ltimes B=v\rtimes B$ for all $B\in\mathbb{B}$ . Likewise, a right central 1v-cell in $\mathbb{B}$ is a 1v-cell $v:A\to \tilde A$ in $\mathbb{B}$ equipped with a vertical pseudonatural transformation $-\rtimes v:-\rtimes A\to -\rtimes\tilde A$ such that $B\ltimes v=B\rtimes v$ for all $B\in\mathbb{B}$ .

A 1v-cell is said to be central if it is both left and right central.

For reader’s convenience, we write down the 2-cell components of the pseudonatural transformations in play. For a horizontal pseudonatural transformation $f\ltimes-:A\ltimes -\to A'\ltimes-$ , a 1h-cell $g:B\to B'$ and a 1v-cell $v:A\to \tilde A$ the 2-cell components $f\ltimes-\vert_g$ and $f\ltimes-\vert_v$ of the oplax transformation structure of $f\ltimes-$ have the form of the left diagrams below (for the lax transformation structure of $f\ltimes-$ the first 2-cell component is different: it is a 2-cell obtained by reading the same upper left diagram from bottom to top). Likewise, the 2-cell components $-\rtimes f\vert_g$ and $-\rtimes f\vert_v$ of the oplax transformation structure of $-\rtimes f:-\rtimes A\to -\rtimes A'$ has the form of the right diagrams below (and the differing 2-cell component for the lax structure is obtained by reading the upper right diagram from bottom to top).

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Similarly, for a 1v-cell $u:B\to \tilde B$ the 2-cell components of oplax transformation structures of $v\ltimes-:A\ltimes -\to \tilde A\ltimes-$ and $-\rtimes v:-\rtimes A\to -\rtimes\tilde A$ have the form as below (and for the lax structures read the respective diagrams from right to left).

(11)

Observe that for a left central 1h-cell f the requirement $f\ltimes B=f\rtimes B$ makes that the image of the pseudodouble functor $-\rtimes B$ at f is related to the images of the pseudodouble functor $A\ltimes-$ at g and v via the corresponding 2-cell components of $f\ltimes-$ , and similarly for other central 1-cells.

Definition 3.3. In a binoidal double category $\mathbb{B}$ a 2-cell in $\mathbb{B}$ , where f and f’ are left central 1h-cells and v and v’ are left central 1v-cells (with respect to horizontal and vertical transformations $f\ltimes-, f'\ltimes$ and $v\ltimes-, v'\ltimes$ , respectively), is said to be left central if for all $B\in\mathbb{B}$ the 2-cells $a\rtimes B$ obey the axioms determining a modification $a\ltimes-$ with 2-cell components $a\ltimes B:=a\rtimes B$ :

and

for any 1h-cell g and any 1v-cell u in $\mathbb{B}$ .

A right central 2-cell is defined similarly. A 2-cell is said to be central if it is both left and right central.

3.2 Centrality and companions

For concluding premonoidality of the underlying bicategory of a premonoidal double category, we will need the following results. They both hold in a binoidal double category. The first one is proved directly.

The right-hand sided version of the claim holds, too.

3.3 Premonoidal double categories

We start by recalling the definition of a premonoidal bicategory from Paquet and Saville (Reference Paquet and Saville2023a) for reader’s convenience.

1. (1) pseudonatural equivalences $\lambda: I\ltimes -\Rightarrow\operatorname {Id}, \rho: -\rtimes I\Rightarrow\operatorname {Id}$ and

   \begin{align*}\begin{array}{c} \alpha_{-,B,C}: (-\rtimes B)\rtimes C\Rightarrow -\rtimes(B\bowtie C)\\\alpha_{A,-,C}: (A\ltimes -)\rtimes C\Rightarrow A\ltimes(-\rtimes C)\\\alpha_{A,B,-}: (A\bowtie B)\ltimes -\Rightarrow A\ltimes(B\ltimes -)\end{array}\end{align*}
   for every $A,B,C\in\mathbb{B}$ , such that all the 1-cell components of these five equivalences are central 1-cells, and $\alpha_{-,B,C}\vert_A=\alpha_{A,-,C}\vert_B=\alpha_{A,B,-}\vert_C$ ;

2. (2) invertible 2-cells $p_{A,B,C,D}, m_{A,B}, l_{A,B}, r_{A,B}$ such that these form modifications in each argument in the form of the modifications $\pi, \mu, \lambda, \rho$ from Reference Gordon, Power and Street Gordon, Power and Street (1995) .

The above data is subject to the same equations between 2-cells as in a monoidal bicategory.

Taking for the starting point the definition of a premonoidal bicategory from Paquet and Saville (Reference Paquet and Saville2023a), one can follow two approaches to introduce a premonoidal double category. One is to add rules to the definition of a premonoidal bicategory for the 1v-cells and to accommodate the rules for 2-cells, so that forgetting the vertical direction one recovers the notion of a premonoidal bicategory.

Another one is to follow Shulman’s recipe from Shulman (Reference Shulman2010). Namely, to consider a premonoidal analogue of a monoidal double category $\mathbb{D}$ from his Shulman (Reference Shulman2010, Definition 2.9), which is a pseudomonoid in a certain 2-category of double categories, and then assuming the existence of companions and conjoints in $\mathbb{D}$ lift isomorphism 1v-cells to equivalence 1h-cells, obtaining that the underlying horizontal bicategory $\mathcal{H}(\mathbb{D})$ is a proper monoidal bicategory.

The latter is a much easier work to do, given that the first way would suppose to work with non-trivial modifications in double categories, while in the Shulman’s way the (vertical) double modifications are trivial, but one still obtains non-trivial (horizontal) modifications $\pi, \mu, \lambda, \rho$ from the definition of a tricategory from Gordon, Power and Street (Reference Gordon, Power and Street1995). (The latter are necessary to have a monoidal bicategory, which is a one object tricategory.) The cost one pays, though, is that this mechanism works only for those double categories whose 1v-cells have companions and conjoints. A synonym for such double categories is framed bicategories, Shulman (Reference Shulman2008, Theorem A.2), and some examples are listed in Shulman (Reference Shulman2008, Section 4.4).

We take the second, Shulman’s approach.

Definition 3.10. Let $\mathbb{B}$ be a binoidal double category. We say that the binoidal structure of $\mathbb{B}$ is associative if there exist invertible vertical strict transformations

\begin{align*}\begin{array}{c}\alpha_{-,B,C}: (-\rtimes B)\rtimes C\Rightarrow -\rtimes(B\bowtie C)\\\alpha_{A,-,C}: (A\ltimes -)\rtimes C\Rightarrow A\ltimes(-\rtimes C)\\\alpha_{A,B,-}: (A\bowtie B)\ltimes -\Rightarrow A\ltimes(B\ltimes -)\end{array} \end{align*}

for every $A,B,C\in\mathbb{B}$ , such that the following is fulfilled

1. (1) 1v-cell components $\alpha_{A,B,C}$ of the above three vertical transformations coincide and are inversely central,

2. (2) and the following four pentagons, expressing equalities of vertical transformations, commute

   

In the pentagons and triangles of a premonoidal double category, the compositions of vertical transformations are the vertical ones, from Lemma 2.6. The 2-cell $(\frac{\alpha_0}{\beta_0})_f$ for two such transformations and a 1h-cell f is given by the vertical composition of the 2-cells $(\alpha_0)_f$ and $(\beta_0)_f$ .

3.4 Other implications in a premonoidal double category

In a premonoidal double category $\mathbb{D}$ , we have

1. (1) pseudodouble functors $A\ltimes -$ and $-\rtimes B$ such that $A\ltimes B=A\rtimes B=:A\bowtie B$ ;

2. (2) a unit object I with invertible vertical strict transformations $\lambda: I\ltimes -\Rightarrow\operatorname {Id}$ and $\rho: -\rtimes I\Rightarrow\operatorname {Id}$ each of whose 1v-cell components are central 1v-cells;

3. (3) invertible vertical strict transformations $\alpha_{-,B,C}, \alpha_{A,-,C},\alpha_{A,B,-}$ for every $A,B,C\in\mathbb{D}$ , such that each 1v-cell component $\alpha_{A,B,C}$ is central and the four pentagons commute;

4. (4) the six triangles commute.

The pseudodouble functors $A\ltimes -$ and $-\rtimes B$ yield functors on the categories $\mathbb{D}_0$ of objects and $\mathbb{D}_1$ of morphisms, and also pseudofunctors $A\ltimes -,-\rtimes B: \mathcal{H}(\mathbb{D})\to \mathcal{H}(\mathbb{D})$ on the underlying horizontal 2-category of $\mathbb{D}$ .

The existence of a distinguished (unit) object $I\in\mathbb{D}$ means that there is a pseudodouble functor $1\to \mathbb{D}$ from the trivial double category so that the image of the single object $*$ is I (and the images of the remaining identity 1h- and 1v-cells and the identity 2-cell on $*$ are horizontal and vertical identities $1_I$ and $1^I$ on I and the identity 2-cell on I, respectively). Then, there are also functors $*\to \mathbb{D}_0$ and $*\to \mathbb{D}_1$ from the trivial category, and a pseudofunctor $*\to \mathcal{H}(\mathbb{D})$ from the trivial 2-category.

Invertible vertical strict transformations are those that have vertically invertible 2-cells $(\alpha_0)_f$ for every 1h-cell f, and they satisfy the axioms (v.l.t.- 1), (v.l.t.- 2) and (v.l.t.- 5) so that the latter axiom is simplified (the 2-cells $\alpha_0^u$ are identities), whereas the compositors in the first axiom and the unitors in the second one are invertible. Equivalently, invertible vertical strict transformations $\omega:F\to G$ between pseudodouble functors consist of two natural transformations $\omega_0:F_0\to G_0$ and $\omega_1:F_1\to G_1$ between the induced functors on the categories of objects and morphisms, such that the axioms (v.l.t.- 1) and (v.l.t.- 2) hold, and the 2-cells $(\alpha_0)_f$ are vertically invertible.

The fact that we work with vertical transformations together with the functor $*\to \mathbb{D}_0$ produces that associative and unital binoidal structure of $\mathbb{D}$ is passed on to an associative and unital binoidal structure of the vertical category $\mathbb{D}_0$ . For the horizontal category $\mathbb{D}_1$ , we consider $1_I$ as the unit object with the rest of a monoidal structure given by the internal category structure of $\mathbb{D}$ .

We may conclude:

We may also prove:

Notational convention. For a premonoidal double category $\mathbb{B}$ whose associativity and unitality constraints are liftable vertical transformations, we will denote by $\underline{\mathcal{H}(\mathbb{B})}$ the underlying bicategory $\mathcal{H}(\mathbb{B})$ equipped with bicategorical pseudonatural equivalences $\hat{\alpha},\hat{\lambda},\hat{\rho}$ induced from $\mathbb{B}$ according to Proposition 2.14. We now know that it is premonoidal.

5.1. Funny functors and funny product for double categories

We denote by Dbl the category of double categories and double functors. The funny product is tied to an inner-hom in which the transformations are not required to obey the naturality condition. Such transformations are called unnatural transformations.

For double categories, we encounter two ways of defining horizontal unnatural transformations (the vertical unnatural transformations are defined then in an analogous way). On one hand, following the logic of unnatural transformations for categories Power and Robinson (Reference Power and Robinson1997) and 2-categories Bourke and Gurski (Reference Bourke and Gurski2017), one can define horizontal unnatural transformations as given merely by their 1h-cell components obeying no axioms. Thus, defined unnatural transformations we will refer to as purely unnatural transformations. Accordingly, modifications between horizontal and analogously defined vertical purely unnatural transformations are given by 2-cell components that are not required to fulfill any axioms. We call them purely funny modifications. Taking for an inner-hom the double category $[\mathbb{A},\mathbb{B}]_f$ of double functors $\mathbb{A}\to \mathbb{B}$ , 1h- and 1v-cells horizontal and vertical purely unnatural transformations, respectively, and purely funny modifications, one obtains the funny product $\mathbb{A}\Box_f\mathbb{B}$ . However, we are interested to build a funny type of a monoidal 2-category of double categories so that pseudomonoids in it would be premonoidal double categories. It will though turn out that in this way we may obtain only premonoidal double categories whose binoidal structure is given by a pair of strict double functors. (The obstacle to obtaining the most general case we will explain in Subsection 5.5.) For that purpose, we will define horizontal unnatural transformations so that they posses square-formed 2-cells that satisfy two axioms. When 1v-cells in these 2-cells are identities, the 2-cells are trivial, so they do retrieve the 2-categorical notion of unnatural transformations. Moreover, instead of taking thus defined unnatural transformations in both directions, for 1v-cells in the inner-hom $[\mathbb{A},\mathbb{B}]$ that we will consider we will take vertical strict transformations.

We define now this second kind of unnatural horizontal transformations of lax double functors between double categories, as well as modifications between them and vertical strict transformations. Such modifications we will call mixed funny modifications. We will then construct a funny product $\mathbb{A}\Box\mathbb{B}$ out of $[\mathbb{A},\mathbb{B}]$ .

1. (1) for every 0-cell A in $\mathbb{A}$ a 1h-cell $\alpha(A)\colon F(A)\to G(A)$ in $\mathbb{B}$ ,

2. (2) for every 1v-cell $u\colon A\to A'$ in $\mathbb{A}$ a 2-cell in $\mathbb{B}$ :

   

so that the following are satisfied (coherence with vertical composition and identity for $\alpha^\bullet$ ): for any composable 1v-cells u and v in $\mathbb{A}$ :

$$ \boldsymbol{(h.u.t.\mbox{-} 1)} \qquad \alpha^{\frac{u}{v}}=\frac{\alpha^u}{\alpha^v}\quad \qquad \text{ and}\quad \qquad \boldsymbol{(h.u.t.\mbox{-} 2)} \qquad \alpha^{1^A}=\operatorname {Id}_{\alpha(A)}.$$

satisfying for every 1v-cell u the axiom (m.hu-vs)

The above definition of unnatural horizontal transformations is coherent with its 2-categorical version indeed. For identity 1v-cells u the 2-cell components $\alpha^u$ of horizontal unnatural transformations are trivial by (h.u.t.- 2) and one recovers unnatural transformations for 2-categories.

Let $[\mathbb{A},\mathbb{B}]$ denote the double category of strict double functors, horizontal unnatural transformations in the sense of Definition 5.1, vertical strict transformations and mixed funny modifications. By $[\mathbb{A},\mathbb{B}]^{lx}$ (resp. $[\mathbb{A},\mathbb{B}]^{ps}$ ), we denote the double category differing from the latter in that its objects are lax (resp. pseudo) double functors $\mathbb{A}\to \mathbb{B}$ . The following result is straightforwardly obtained, it is a mixed funny version of Femić (Reference Femić2023, Proposition 3.3). The reader may consult Table 7 to keep track of the relevant axioms (minding the fact that the horizontal transformations are now unnatural and that the vertical transformations are now strict).

1. (1) two families of lax double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}\quad \text{ and}\quad (B,-)\colon\mathbb{A}\to \mathbb{C}$ for objects $A\in\mathbb{A}, B\in\mathbb{B}$ , such that $H(A,-)=(-, A), H(-, B)=(B,-)$ and $(-,A)\vert_B=(B,-)\vert_A=(B,A)$ , and

2. (2) two families of 2-cells

   

and a family of (identity) 2-cells

in $\mathbb{C}$ determined by all 1h-cells $K\colon A\to A'$ and 1v-cells $U\colon A\to \tilde A$ in $\mathbb{A}$ , and 1h-cells $k\colon B\to B'$ and 1v-cells $u\colon B\to \tilde B$ in $\mathbb{B}$ , which satisfy the following 11 axioms from Proposition 1:

(( $u,1_A$ )), (( $1_B,U$ )), (( $1^B,K$ )), (( $k,1^A$ )),

((u, K’K)), ((k’k, U)), (( $\frac{u}{u'}, K$ )), (( $k,\frac{U}{U'}$ )),

$(u,U)=\operatorname {Id}$ , (u,U)-l-nat((u,U)-l-nat), (u,U)-r-nat((u,U)-r-nat).

Proposition 5.4. A double functor $\mathcal{F}\colon\mathbb{A}\to [\mathbb{B}, \mathbb{C}]$ is a lax double functor $\mathcal{F}\colon\mathbb{A}\to [\mathbb{B}, \mathbb{C}]^{lx}$ in whose data the double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}$ and $(B,-)\colon\mathbb{A}\to \mathbb{C}$ are strict. In this case, the axioms (( $u,1_A$ )), (( $1_B,U$ )), ((u, K’K)), ((k’k, U)) simplify into the expressions:

$$(u,1_A)=\operatorname {Id}^{(u,A)}, \,\, (1_B,U)=\operatorname {Id}^{(B,U)}, \,\, (u,K'K)=[(u,K)\vert(u,K')], \,\, (k'k,U)=[(k,U)\vert(k',U)].$$

Analogously to Proposition 5.3 one characterizes a pseudodouble functor $\mathcal{F}\colon\mathbb{A}\to [\mathbb{B}, \mathbb{C}]^{ps}$ . It satisfies formally same 11 axioms: with the only change that the structure 2-cells of the pseudodouble functors $(-,A)\colon\mathbb{B}\to \mathbb{C}$ and $(B,-)\colon\mathbb{A}\to \mathbb{C}$ are invertible.

A pseudodouble funny functor $\mathbb{A}\times\mathbb{B}\to \mathbb{C}$ is analogously defined.

A funny functor $\mathbb{A}\times\mathbb{B}\to \mathbb{C}$ between double categories (or a double funny functor) is a pair of double functors $(-,A):\mathbb{B}\to \mathbb{C}$ and $(B,-):\mathbb{A}\to \mathbb{C}$ satisfying the conditions of Proposition 5.4.

Strictly speaking the above funny functors we should call mixed funny functors, as in the vertical direction they are not determined by unnatural transformations. We will use the abbreviated version of the name. We proceed to construct a funny product of double categories $\mathbb{A}$ and $\mathbb{B}$ . One obtains a strict funny product $\mathbb{A}\Box\mathbb{B}$ by considering the inner-hom $[\mathbb{B},\mathbb{C}]$ , while a lax version $\mathbb{A}\Box^{lx}\mathbb{B}$ is obtained by considering the inner-hom $[\mathbb{B},\mathbb{C}]^{lx}$ (and similarly for $\mathbb{A}\Box^{ps}\mathbb{B}$ ). We will show now the construction for the first one (the strict one), as it is this one that is going to give a monoidal category structure (of funny type) to the category of double categories.

To obtain a funny product $\mathbb{A}\Box\mathbb{B}$ , we suppose there is a left adjoint $-\Box \mathbb{B}$ to $[\mathbb{B},-]$ . For this set $\mathbb{C}=\mathbb{A}\times\mathbb{B}$ , consider the unit of the adjunction $E: \mathbb{A}\to [\mathbb{B},\mathbb{A}\times\mathbb{B}]$ and read off the structure of the image double category $E(\mathbb{A})(\mathbb{B})$ in $\mathbb{A}\times\mathbb{B}$ . It is a double category $\mathbb{A}\Box\mathbb{B}$ generated by certain pairs $(x,y)=:x\Box y$ , where $x\in\mathbb{A}$ and $y\in\mathbb{B}$ are 0-, 1h-, 1v- or 2-cells, that we also get as images of the above two double functors $(-,A)$ and $(B,-)$ (from the end of Definition 5.5). Although we do not know the above pair of functors nor E, knowing that $\mathbb{A}\Box\mathbb{B}$ is generated as the image by double functors $(-,A)$ and $(B,-)$ , gives us the hint on how to define it.

Namely, we define a funny product $\mathbb{A}\Box\mathbb{B}$ by the following generators and relations:

objects: $A\Box B$ for objects $A\in\mathbb{A}, B\in\mathbb{B}$ ;

1h-cells: $A\Box k, K\Box B$ , where k is a 1h-cell in $\mathbb{B}$ and K a 1h-cell in $\mathbb{A}$ ;

1v-cells: $A\Box u, U\Box B$ and vertical compositions of such obeying the following rules:

(14) \begin{equation} \frac{A\Box u}{A\Box u'}=A\Box \frac{u}{u'}, \quad \frac{U\Box B}{U'\Box B}=\frac{U}{U'}\Box B, \quad A\Box 1^B=1^{A\Box B}=1^A\Box B\end{equation}

where u,u’ are 1v-cells of $\mathbb{B}$ and U,U’ 1v-cells of $\mathbb{A}$ ;

2-cells: $A\Box\omega, \zeta\Box B$ , where $\omega$ is a 2-cell in $\mathbb{B}$ and $\zeta$ is a 2-cell in $\mathbb{A}$ , further 2-cells

(15)

subject to the eleven relations:

(16) \begin{equation} K\Box 1^B=Id_{K\Box B}, \,\, 1^A\Box k=Id_{A\Box k}, \,\, K\Box \frac{u}{u'}=\frac{K\Box u}{K\Box u'}, \,\,\frac{U}{U'}\Box k=\frac{U\Box k}{U'\Box k},\end{equation}

(17) \begin{equation} 1_A\Box u=\operatorname {Id}^{A\Box u}, \,\, U\Box 1_B=\operatorname {Id}^{U\Box B}, \,\, K'K\Box u=[K\Box u\vert K'\Box u], \,\, U\Box k'k=[U\Box k\vert U\Box k'];\end{equation}

(18) \begin{equation} U\Box u=\operatorname {Id}, \,\,\,\, \frac{A\Box\omega}{U\Box l}=\frac{U\Box k}{\tilde A\Box\omega}, \,\,\,\, \frac{K\Box u}{\zeta\Box\tilde B}=\frac{\zeta\Box B}{L\Box u};\end{equation}

four equations holding from the strictness of double functors $(-,A)$ and $(B,-)$ :

(19) \begin{equation} (A\Box k')(A\Box k)= A\Box (k' k), \quad (K'\Box B)(K\Box B)=(K' K)\Box B\end{equation}

$$1_{A\Box B}= A\Box 1_B,\quad 1_{A\Box B}= 1_A\Box B$$

and the following ones:

(20) \begin{equation} A\Box \omega'\omega=[A\Box\omega \vert A \Box \omega'], \quad \zeta'\zeta\Box B=[\zeta\Box B \vert\zeta' \Box B]\end{equation}

(21) \begin{equation} A\Box\frac{\omega}{\omega'}=\frac{A\Box\omega}{A\Box\omega'}, \quad \frac{\zeta}{\zeta'}\Box B=\frac{\zeta\Box B}{\zeta'\Box B},\end{equation}

(22) \begin{equation} A\Box\operatorname {Id}_k=\operatorname {Id}_{A\Box k}, \quad \operatorname {Id}_K\Box B=\operatorname {Id}_{K\Box B}, \quad A\Box\operatorname {Id}^u=\operatorname {Id}^{A\Box u}, \quad \operatorname {Id}^U\Box B=\operatorname {Id}^{U\Box B} .\end{equation}

The source and target functors s,t on $\mathbb{A}\Box\mathbb{B}$ are defined as in the double category $\mathbb{A}\times\mathbb{B}$ , the composition functor c is defined by horizontal juxtaposition of the corresponding 2-cells, and the unit functor i is defined on generators as follows:

$$i(A\Box B)=1_{A\Box B}, \, i(A\Box v)=1^A\Box v (=Id^{A\Box v}) \quad \text{and} \quad i(U\Box B)=U\Box 1^B (=Id^{U\Box B}).$$

Now it is straightforward to see that $-\Box\mathbb{B}: Dbl\to Dbl$ (resp. $\mathbb{B}\Box-: Dbl\to Dbl$ ) and $[\mathbb{B}, -]$ are double functors. Then by construction, we have bijections

$$Dbl(\mathbb{A}\Box\mathbb{B},\mathbb{C})\cong f\mbox{-} Dbl(\mathbb{A}\times\mathbb{B},\mathbb{C})\cong Dbl(\mathbb{A}, [\mathbb{B}, \mathbb{C}]),$$

where $Dbl(\mathbb{A}, \mathbb{B})$ denotes a set of double functors $\mathbb{A}\to \mathbb{B}$ and $f\mbox{-} Dbl(\mathbb{A}\times\mathbb{B},\mathbb{C})$ is the set of double funny functors.

It is readily seen that the above bijections are natural in all variables and that the product $-\Box-$ is symmetric. We defer the proof that $(Dbl, \Box)$ is a (symmetric) monoidal category for Subsection 5.3, Theorem 5.30, then it will be a biclosed symmetric monoidal category.

In the general case, where $*$ stands for any kind of funny functors (be it strict, lax, colax, or pseudo), funny functors of type $*$ are by construction such that there is a bijection

(23) \begin{equation}f\mbox{-}\operatorname {Fun}^*(\mathbb{A}\times\mathbb{B},\mathbb{C})\cong \operatorname {Fun}^*(\mathbb{A}, [\mathbb{B},\mathbb{C}]^*).\end{equation}

Here, $f\mbox{-}\operatorname {Fun}^*(\mathbb{A}, \mathbb{B})$ and $\operatorname {Fun}^*(\mathbb{A}, \mathbb{B})$ stand for the sets of double funny functors and double functors of type $*$ , respectively, and $[\mathbb{B}, \mathbb{C}]^*$ is the $*$ -typed double functor version of $[\mathbb{B}, \mathbb{C}]$ . Also, the induced funny product fits the bijection

$$Dbl(\mathbb{A}\Box^*\mathbb{B},\mathbb{C})\cong\operatorname {Fun}^*(\mathbb{A},[\mathbb{B},\mathbb{C}]^*)$$

so that one gets

(24) \begin{equation} Dbl(\mathbb{A}\Box^*\mathbb{B},\mathbb{C})\cong f\mbox{-}\operatorname {Fun}^*(\mathbb{A}\times\mathbb{B},\mathbb{C}).\end{equation}

We state for the record that $[-,-]^*$ and $-\Box^*-$ are functors only on Dbl and $Dbl_{ps}$ , the categories of double categories with strict or pseudodouble functors, but they are not functors on $Dbl_{lx}$ nor $Dbl_{clx}$ (where morphisms are lax or colax double functors). (For the reason of this failure, see Femić (Reference Femić2023, Section 3.1).)

Given that $\mathbb{A}\Box^*\mathbb{B}$ is defined by generators and relations on $\mathbb{A}\times\mathbb{B}$ , there is a double funny functor

(25) \begin{equation} J^*\colon\mathbb{A}\times\mathbb{B}\to \mathbb{A}\Box^*\mathbb{B}\end{equation}

of type $*$ given by $J^*(-, B)(a)=a\Box B, J^*(A,-)(b)=A\Box b$ for cells a in $\mathbb{A}$ and b in $\mathbb{B}$ and with unique 2-cells $U\Box k, K\Box u$ satisfying the eleven axioms.

We end this subsection by giving another characterization of lax double funny functors analogous to that of lax double quasi-functors from Femić (Reference Femić2023, Proposition 3.3) (see Proposition 1).

1. (1) $H\colon \mathbb{A}\times\mathbb{B}\to \mathbb{C}$ is a lax double funny functor,

2. (2) there are two families of lax double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}\quad \text{ and}\quad (B,-)\colon\mathbb{A}\to \mathbb{C}$ for objects $A\in\mathbb{A}, B\in\mathbb{B}$ , such that $(-,A)\vert_B=(B,-)\vert_A=(B,A)$ , and the following hold:

   1. (i) $(-,K)\colon (-,A)\to (-,A')$ is a horizontal unnatural transformation for each 1h-cell $K\colon A\to A'$ , $(-,U)\colon (-,A)\to (-,\tilde A)$ is a vertical strict transformation for each 1v-cell $U\colon A\to \tilde A$ in $\mathbb{A}$ , $(-,\zeta)$ is a mixed funny modification with respect to horizontally unnatural and vertically strict transformations for each 2-cell $\zeta$ in $\mathbb{A}$ , and the following coincide:

      $$(B,-)\vert_K=(-,K)\vert_B, \,\,\,\, (B,-)\vert_U=(-,U)\vert_B, \,\,\,\, (B,-)\vert_\zeta=(-,\zeta)\vert_B;$$

   2. (ii) $(k,-)\colon (B,-)\to (B', -)$ is a horizontal unnatural transformation for each 1h-cell $k\colon B\to B'$ , $(u,-)\colon (B,-)\to (\tilde B,-)$ is a vertical strict transformation for each 1v-cell $u\colon B\to \tilde B$ in $\mathbb{B}$ , $(\omega,-)$ is a mixed funny modification with respect to horizontally unnatural and vertically strict transformations for each 2-cell $\omega$ in $\mathbb{B}$ , and the following coincide:

      $$(-,A)\vert_k=(k,-)\vert_A, \,\,\,\, (-,A)\vert_u=(u,-)\vert_A, \,\,\,\, (-,A)\vert_\omega=(\omega,-)\vert_A;$$

   3. (iii) for 1h-cells K,k and 1v-cells U,u the following 2-cell components of the respective transformations coincide: $(-,K)\vert_u=(u,-)\vert_K$ and $(-,U)\vert_k=(k,-)\vert_U$ .

The pseudo (double funny functor) version of the above proposition for $\mathbb{A}=\mathbb{B}=\mathbb{C}$ provides a characterization of a binoidal structure coming from a pseudodouble funny functor:

1. • $\mathbb{B}$ is binoidal;

2. • there are 1-cells

   $$-\rtimes k\vert_A:=A\ltimes-\vert_k, \,\,\, -\rtimes u\vert_A:=A\ltimes-\vert_u \qquad K\ltimes-\vert_B:=-\rtimes B\vert_K, \,\,\, U\ltimes-\vert_B:=-\rtimes B\vert_U; $$

3. • there are 2-cells $K\ltimes-\vert_u=-\rtimes u\vert_K$ functorial both in k and in u, and $U\ltimes-\vert_k=-\rtimes k\vert_U$ functorial both in U and in k;

4. • all 1v-cells u are central in $\mathbb{B}$ via vertical strict transformations $u\ltimes-$ and $-\rtimes u$ ;

5. • there are 2-cells $-\rtimes \omega\vert_A:=A\ltimes-\vert_\omega$ and $\zeta\ltimes-\vert_B:=-\rtimes B\vert_\zeta$ natural with respect to 1v-cells (in the sense of (m.hu-vs)),

for all objects A,B, 1h-cells K,k, 1v-cells U,u and 2-cells $\zeta, \omega$ in $\mathbb{B}$ .

5.1.1. Purely funny product

All the results of the above part of Subsection 5.1 have their counterparts for the funny product obtained from the inner-hom $[\mathbb{A},\mathbb{B}]_f$ whose 1h- and 1v-cells are horizontal and vertical purely unnatural transformations. In this subsection, we record these results. The obtained funny product we denote by $\mathbb{A}\Box_f\mathbb{B}$ and refer to it as a purely funny product.

The “pure” version of Proposition 5.3 states that a lax double functor $\mathcal{F}\colon\mathbb{A}\to [\mathbb{B}, \mathbb{C}]_f^{lx}$ of double categories consists merely of two families of lax double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}$ and $(B,-)\colon\mathbb{A}\to \mathbb{C}$ for objects $A\in\mathbb{A}, B\in\mathbb{B}$ , such that $H(A,-)=(-, A), H(-, B)=(B,-)$ and $(-,A)\vert_B=(B,-)\vert_A=(B,A)$ . This short data we may call purely (lax) funny functor $\mathbb{A}\times\mathbb{B}\to \mathbb{C}$ of double categories.

Namely, by substituting strict vertical transformations by unnatural ones, the axioms $(u,u)=\operatorname {Id}$ , ((u,U)-l-nat) and ((u,U)-r-nat) are eliminated. Furthermore, by using purely unnatural transformations, one eliminates square-formed 2-cells and $2+2$ axioms in both directions (h.u.t.- 1), (h.u.t.- 2) and their vertical counterparts) stemming from both variables in a (mixed) funny functor. In total, 11 axioms from the definition of a mixed funny functor are omitted. Consequently, 11 axioms from the definition of the funny product $\mathbb{A}\Box\mathbb{B}$ are omitted to obtain the purely funny product $\mathbb{A}\Box_f\mathbb{B}$ . It will have the same generators on objects and 1-cells, but its only generating 2-cells will be $A\Box_f\omega$ and $\zeta\Box_f B$ .

The “pure” versions of bijections (23) and (24) read:

$$pf\mbox{-}\operatorname {Fun}^*(\mathbb{A}\times\mathbb{B},\mathbb{C})\cong \operatorname {Fun}^*(\mathbb{A}, [\mathbb{B},\mathbb{C}]_f^*) \qquad \text{and}\qquad Dbl(\mathbb{A}\Box_f^*\mathbb{B},\mathbb{C})\cong pf\mbox{-}\operatorname {Fun}^*(\mathbb{A}\times\mathbb{B},\mathbb{C}).$$

Finally, Proposition 5.8 in its “pure” version states:

Proposition 5.11 Let $\mathbb{A},\mathbb{B},\mathbb{C}$ be double categories. The following are equivalent:

1. (1) $H\colon \mathbb{A}\times\mathbb{B}\to \mathbb{C}$ is a lax double purely funny functor,

2. (2) there are two families of lax double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}\quad \text{ and}\quad (B,-)\colon\mathbb{A}\to \mathbb{C}$ for objects $A\in\mathbb{A}, B\in\mathbb{B}$ , such that $(-,A)\vert_B=(B,-)\vert_A=(B,A)$ , and the following hold:

   1. (i) $(-,K)\colon (-,A)\to (-,A')$ is a horizontal purely unnatural transformation for each 1h-cell $K\colon A\to A'$ , $(-,U)\colon (-,A)\to (-,\tilde A)$ is a vertical purely unnatural transformation for each 1v-cell $U\colon A\to \tilde A$ in $\mathbb{A}$ , $(-,\zeta)$ is a purely funny modification for each 2-cell $\zeta$ in $\mathbb{A}$ , and the following coincide:

      $$(B,-)\vert_K=(-,K)\vert_B, \,\,\,\, (B,-)\vert_U=(-,U)\vert_B, \,\,\,\, (B,-)\vert_\zeta=(-,\zeta)\vert_B;$$

   2. (ii) $(k,-)\colon (B,-)\to (B', -)$ is a horizontal purely unnatural transformation for each 1h-cell $k\colon B\to B'$ , $(u,-)\colon (B,-)\to (\tilde B,-)$ is a vertical purely unnatural transformation for each 1v-cell $u\colon B\to \tilde B$ in $\mathbb{B}$ , $(\omega,-)$ is a purely funny modification for each 2-cell $\omega$ in $\mathbb{B}$ , and the following coincide:

      $$(-,A)\vert_k=(k,-)\vert_A, \,\,\,\, (-,A)\vert_u=(u,-)\vert_A, \,\,\,\, (-,A)\vert_\omega=(\omega,-)\vert_A.$$

The meaning of points (i) and (ii) in the above proposition is that the component 1-cells of horizontal and vertical purely unnatural transformations and the component 2-cells of purely funny modifications coincide with the corresponding images of the lax double functors $(-,A)\colon\mathbb{B}\to \mathbb{C}$ and $(B,-)\colon\mathbb{A}\to \mathbb{C}$ . In particular, the pseudo (double purely funny) version of the proposition with $\mathbb{A}=\mathbb{B}=\mathbb{C}$ implies that in a premonoidal double category in which the binoidal structure comes from a pseudodouble purely funny functor, we have that there exist 1- cells $-\rtimes k\vert_A:=A\ltimes-\vert_k, \,\,\,\, -\rtimes u\vert_A:=A\ltimes-\vert_u, \quad K\ltimes-\vert_B:=-\rtimes B\vert_K, \,\,\,\, U\ltimes-\vert_B:=-\rtimes B\vert_U$ and 2-cells $-\rtimes \omega\vert_A:=A\ltimes-\vert_\omega$ and $\zeta\ltimes-\vert_B:=-\rtimes B\vert_\zeta$ for which no further laws are required. Then we may conclude:

Corollary 5.12. Any binoidal structure in a double category $\mathbb{B}$ is given by a pseudodouble purely funny functor $H:\mathbb{B}\times\mathbb{B}\to \mathbb{B}$ .

5.2 Relation to the funny product for 2-categories

Funny product for 2-categories was studied in Bourke and Gurski (Reference Bourke and Gurski2017, Section 2). Its construction follows an analogous process as our construction of the funny product for double categories, starting from an inner-hom that there was denoted by $[\mathcal{A},\mathcal{B}]_f$ for 2-categories $\mathcal{A},\mathcal{B}$ . The inner-hom $[\mathcal{A},\mathcal{B}]_f$ has for objects 2-functors, for 1-cells transformations (consisting only of 1-cell components subject to no axioms), and for 2-cells modifications (consisting only of 2-cell components subject to no axioms). The obtained funny tensor product there was denoted by $\mathcal{A}\star\mathcal{B}$ , it satisfies the universal property

$$2\mbox{-}\operatorname {Cat}(\mathcal{A}\star\mathcal{B},\mathcal{C})\cong 2\mbox{-}\operatorname {Cat}(\mathcal{A},[\mathcal{B},\mathcal{C}]_f)$$

and gives a symmetric closed monoidal structure on the category $2\mbox{-}\operatorname {Cat}$ of 2-categories. Analogously as in Subsection 5.1.1, a funny functor $\mathcal{A}\star\mathcal{B}\to \mathcal{C}$ between 2-categories is given merely by a pair of families of 2-functors $(-,A):\mathcal{B}\to \mathcal{C}$ and $(B,-):\mathcal{A}\to \mathcal{C}$ for $A\in\mathcal{A},B\in\mathcal{B}$ satisfying $(-,A)_B=(B,-)_A=(B,A)$ . An explicit description of $\mathcal{A}\star\mathcal{B}$ can be deduced from our description of $\mathcal{A}\Box_f\mathcal{B}$ for double categories by considering all vertical 1-cells to be identities. Thus, it is given by the following generators and relations:

objects: $A\star B$ for objects $A\in\mathcal{A}, B\in\mathcal{B}$ ;

1-cells: $A\star k, K\star B$ , where k is a 1-cell in $\mathcal{B}$ and K a 1-cell in $\mathcal{A}$ ;

2-cells: $A\star\omega, \zeta\star B$ , where $\omega$ is a 2-cell in $\mathbb{B}$ and $\zeta$ a 2-cell in $\mathcal{A}$ ;

four equations from the strictness of double functors $(-,A)$ and $(B,-)$ :

$$(A\star k')(A\star k)= A\star (k' k), \quad (K'\star B)(K\star B)=(K' K)\star B$$

$$1_{A\star B}= A\star 1_B,\quad 1_{A\star B}= 1_A\star B$$

and the following ones:

$$A\star \omega'\omega=[A\star\omega \vert A \star \omega'], \quad \zeta'\zeta\star B=[\zeta\star B \vert\zeta' \star B]$$

$$A\star\frac{\omega}{\omega'}=\frac{A\star\omega}{A\star\omega'}, \quad \frac{\zeta}{\zeta'}\star B=\frac{\zeta\star B}{\zeta'\star B},$$

$$A\star\operatorname {Id}_k=\operatorname {Id}_{A\star k}, \quad \operatorname {Id}_K\star B=\operatorname {Id}_{K\star B}.$$

Likewise, the funny product on morphisms, i.e., on 2-functors, is defined by $(F\star G)(a\star b)=F(a)\star G(b)$ for sensible cells $a\star b\in\mathcal{A}\star\mathcal{B}$ . This way we have a functor $-\star-:2\mbox{-}\operatorname {Cat}\times 2\mbox{-}\operatorname {Cat}\to 2\mbox{-}\operatorname {Cat}$ . Inspecting the above generators of $\mathcal{A}\star\mathcal{B}$ we see that $F\star G$ is indeed given by a pair of families of 2-functors $F\star G(B), F(A)\star G$ for objects $A\in\mathcal{A}, B\in\mathcal{B}$ .

Our purely funny product of double categories $\mathbb{A}\Box_f\mathbb{B}$ is such that there is a monoidal embedding $(2\mbox{-}\operatorname {Cat}, \star)\hookrightarrow(Dbl, \Box_f)$ .

Consider the 2-category of 2-categories $(2\mbox{-}\operatorname {Cat})_2$ and the 2-category $Dbl_\bullet$ of double categories, double functors and some $\bullet$ -type of vertical transformations.

By the generators of the funny product 2-category $(2\mbox{-}\operatorname {Cat})_2\star(2\mbox{-}\operatorname {Cat})_2$ , for 2-functors F,G, i.e. 1-cells in $(2\mbox{-}\operatorname {Cat})_2$ , there is no valid 1-cell $F\star G$ living in $(2\mbox{-}\operatorname {Cat})_2\star(2\mbox{-}\operatorname {Cat})_2$ . Rather, there are only 1-cells of the form $F\star\mathcal{B}$ and $\mathcal{A}\star G$ for 2-categories $\mathcal{A}$ and $\mathcal{B}$ .

Similarly, in the funny product 2-category $Dbl_\bullet\star Dbl_\bullet$ if we set $\mathbb{A}\star\mathbb{B}:=\mathbb{A}\Box_f\mathbb{B}$ on the objects, the purely funny product of double categories, the only valid 1-cells are of the form $F\star\mathbb{B}:=F\Box_f \operatorname {Id}_\mathbb{B}: \mathbb{A}\Box_f\mathbb{B}\to \mathbb{A}'\Box_f\mathbb{B}$ and $\mathbb{A}\star G:=\operatorname {Id}_\mathbb{A}\Box_f G:\mathbb{A}\Box_f\mathbb{B}\to \mathbb{A}\Box_f\mathbb{B}'$ , where the latter are defined similarly as in Remark 5.6. Namely, the former is given by the pairs of double functors $F(A)\Box_f-, F(-)\Box_f B$ , and the latter by the pairs $A\Box_f G(-), -\Box_f G(B)$ .

However, had we set $\mathbb{A}\star\mathbb{B}:=\mathbb{A}\Box\mathbb{B}$ (the mixed funny product of double categories) on the objects in the funny product 2-category $Dbl_\bullet\star Dbl_\bullet$ , the 1-cells would be of the form $F\star\mathbb{B}:=F\Box\operatorname {Id}_\mathbb{B}$ and $\mathbb{A}\star G:=\operatorname {Id}_\mathbb{A}\Box G$ , defined in Remark 5.6. That is, the former is given by the pairs of double functors $F(A)\Box-, F(-)\Box B$ and three families of 2-cells $(u,K):=F(K)\Box u, \, \, (k,U):=F(k)\Box U, \,\, (u,U):=F(U)\Box u$ obeying the eleven axioms (16) – (18), and the latter is given by the pairs $A\Box G(-), -\Box G(B)$ and three families of 2-cells $(u,K):=K\Box G(u), \, \, (k,U):=k\Box G(U), \,\, (u,U):=U\Box G(u)$ satisfying the eleven axioms.

5.3. Multicategories induced by funny functors

It was shown in Hermida (Reference Hermida2000, Theorem 9.8) that monoidal categories are in 1-1 correspondence with representable multicategories. We remind the reader of the notion of a representable multicategory.

1. • a collection of objects,

2. • for each list of objects $a_1,...,a_n$ for $n\geq 0$ and an object b, a set $\mathcal{M}_n(a_1,...,a_n;b)$ ,

3. • for each object a an element $1_a\in\mathcal{M}_1(a;a)$ ,

4. • for all lists $\overline{a_i}, i=1,...,n$ , and objects $b_1,...,b_n$ and c a function, called substitution:

   $$\mathcal{M}_n(b_1,...,b_n,c)\times\Pi_{j=1}^{n}\mathcal{M}_{k_i}(\overline{a_i};b_i)\to \mathcal{M}_{\sum_{i=1}^n k_i}(\overline{a_1},...,\overline{a_n};\;c)$$
   $$(g, f_1,...,f_n)\mapsto g\circ(f_1,...,f_n)$$
   satisfying a natural associativity and two identity axioms. Here $\overline{a_i}$ is a short annotation for a list of objects, and the elements of $\mathcal{M}_n(a_1,...,a_n;b)$ are called multimaps.

Definition 5.15. A multicategory $\mathcal{M}$ is said to be representable if for any list of objects $\overline a=a_1,..,a_n$ there exists an object $m(a_1,..,a_n)$ and a multimap $j_{\overline a}:a_1,..,a_n\to m(a_1,..,a_n)$ that for every object c and lists $\overline x, \overline b$ (of lengths k and l, respectively) induces bijections

$$\mathcal{M}_1(m(a_1,..,a_n);\;c)\to \mathcal{M}_n(a_1,..,a_n;\;c)$$

and

$$\mathcal{M}_{k+1+l}(\overline x,m(a_1,..,a_n),\overline b;\;c)\to \mathcal{M}_{k+n+l}(\overline x,a_1,..,a_n,\overline b;\;c).$$

We also recall when a multicategory is closed.

We are going to show that (both purely and mixed) funny functors make multimaps for a multicategory. Moreover, these multicategories will be represented by their respective funny types of product. In the first six subsections of Subsection 5.3, we will study the mixed funny case leading to the (mixed) funny monoidal product $\Box$ , omitting the adjective mixed for simplicity. In Subsection 5.3.8, we will briefly record the corresponding results for the purely funny case of $\Box_f$ . The constructions that follow are a funny version of the construction that we carried out in Femić (Reference Femić2024) with quasi-functors for double categories.

5.3.1. Ternary and n-ary (lax) double funny functors: constructing a multicategory

Analogously as we do in the next definition for double categories, one can define funny functors of more than two variables for categories and 2-categories.

Definition 5.17. A lax double funny functor of n-variables $H:\mathbb{A}_1\times...\times\mathbb{A}_n\to \mathbb{C}$ for $n\geq 2$ consists of binary lax double funny functors

$$H(A_1,...,A_{i-1},\, -,\, A_{i+1},...,A_{j-1}, \, -,\, A_{j+1},..., A_n): \mathbb{A}_i\times\mathbb{A}_j\to \mathbb{C}$$

for all $i<j$ and all choices of objects $A_l\in\mathbb{A}_l, l=1,...,n$ , which agree on objects as lax double functors of 1-variable (that is, they give unambiguous lax double functors

$$H(A_1,...,A_{i-1},\, -,\,A_{i+1},..., A_n): \mathbb{A}_i\to \mathbb{C}$$

for all $i=1,...,n$ ), and satisfy the axioms:

(u,v,h)

for $(u,v,h):(A,B,C)\to (\tilde A,\tilde B,C')$ in $\mathbb{A}_i\times\mathbb{A}_j\times\mathbb{A}_k$ , whereby we omit writing the rest of the $n\mbox{-} 3$ variables, and 2 similar axioms: (u,g,z) for $(u,g,z):(A,B,C)\to (\tilde A,B',\tilde C)$ and (f,v,z) for $(f,v,z):(A,B,C)\to (A',\tilde B,C')$ , where f,g,h are 1h-cells and u,v,z are 1v-cells, as usual.

We illustrate the condition that the underlying funny functors of 2-variables give unambiguous lax double functors of 1-variables in the example of ternary funny functors $H:\mathbb{A}\times\mathbb{B}\times\mathbb{C}\to \mathbb{D}$ . This means:

(26) \begin{equation} H(A,-,-)\vert_B=H(-,B,-)\vert_A, H(A,-,-)\vert_C=H(-,-,C)\vert_A, H(-,B,-)\vert_C=H(-,-,C)\vert_B\end{equation}

for $A\in\mathbb{A},B\in\mathbb{B},C\in\mathbb{C}$ . The above definition can be restated so that an n-ary lax double funny functor consists of lax double funny functors of $(n\mbox{-} 1)$ -variables (and hence also of k-variables for all $2\leq k<n$ ) that for any choice of three variables satisfy the condition (26). We will sometimes refer to the latter condition as the ternary funny property (with functorialities in the third, second and first variable, respectively.

It is clear from the definition that any kind of double funny functors $H:\mathbb{A}_1\times...\times\mathbb{A}_n\to \mathbb{C}$ for $n\geq 2$ of double categories are not double functors defined on $\mathbb{A}_1\times...\times\mathbb{A}_n$ . Loosely speaking, they are only defined on cells $(a_1,...,a_n)\in\mathbb{A}_1\times...\times\mathbb{A}_n$ whereby either all $a_i$ are objects, or at most two of them are simultaneously 1-cells (in which case one is a 1h-cell and another a 1v-cell) or a single $a_i$ is a 2-cell, whereas the rest are objects.

Let us now introduce the sets of multimaps for our multicategory of funny type for double categories. For the class of nullary maps $\mathcal{M}_0^*(-;\mathbb{A})$ , we set the set of objects of $\mathbb{A}$ , and for unary maps we set $\mathcal{M}_1^*(\mathbb{A};\;\mathbb{B})=\operatorname {Fun}^*(\mathbb{A},\mathbb{B})$ . For the class of n-ary multimaps, we set $\mathcal{M}_n^*(\mathbb{A}_1,...,\mathbb{A}_n;\mathbb{C})=f_n\mbox{-}\operatorname {Fun}^*(\mathbb{A}_1\times...\times\mathbb{A}_n,\mathbb{C})$ , the set of double funny functors of type $*$ of n-variables.