Pseudofunctors of 2-variables and Gray tensor product of bicategories

Question

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be bicategories. We say a 'pseudofunctors of 2-variables' consists of two families of pseufunctors

1. Fix $A\in obj\mathcal{A}$, we have a pseudofunctor $f(A,-): \mathcal{B}\to \mathcal{C}$.

2. Fix $B\in obj\mathcal{B}$, we have a psudofuncotr $f(-,B): \mathcal{A}\to \mathcal{C}$.

3. $f(A,-)(B)=f(-,B)(A):=f(A,B)$ for all $A\in obj\mathcal{A}, B\in\mathcal{B}$.

4. for all $\alpha: A\to A',\beta:B\to B'$, an invertible 2-cell $f(\alpha,\beta)$ between $$ f(A,B)\xrightarrow{f(A, \beta)}f(A,B')\xrightarrow{f(\alpha,B')} f(A',B'),\\ f(F,B)\xrightarrow{f(\alpha, B)}f(A',B)\xrightarrow{f(A',\beta)}f(A',B'). $$

These two families and $f(\alpha,\beta)$ satisfy coherence conditions similar to that of quasi-functor of two variables introduced by Gray ("Adjointness for 2-Categories", page 56), but droping the first requirement on identities, which I shall not spell out here; see Gordon-Power-Street in "Coherence for tricategory", where it is called cubical functors (so remove the line "which is an identity when either $\alpha$ or $\beta$ is an identity).

Notice that if $\mathcal{A}=\mathcal{B}=*$, the trivial bicategory with one objects, one 1-cell, one 2-cell, then the above data should be equivalent to two monads in $\mathcal{C}$ on the same object satisfying a distributive law.

Question:

Does there exist a funny tensor product $\mathcal{A}\otimes \mathcal{B}$ such that the above data is the same as a single pseudofunctor $\mathcal{A}\otimes \mathcal{B}\to \mathcal{C}$? (Gray tensor product of bicategories?)

Answer 1

Since this question was written, there is a paper 2-dimensional bifunctor theorems and distributive laws by Faul–Manuell–Siqueira based on this idea. In particular, they prove a "bifunctor theorem" for lax functors between 2-categories (Theorem 3.2), and relate it to certain tensor products of 2-categories (see Theorem 4.10, the end of §4, and Theorem 5.3). Also relevant is the paper Bifunctor Theorem and strictification tensor product for double categories with lax double functors of Femić.