## Background

There is a bicategory where the objects are categories, the 1-morphisms are profunctors, and the 2-morphisms are morphisms of profunctors. The non-obvious part of this assertion is that profunctors admit a "good" (i.e., coherently associative) composition. The way one sees this is to use the fact that the Yoneda embedding is the free cocompletion, from which we may identify profunctors from $\mathcal{C}$ to $\mathcal{D}$ with cocontinuous functors from $\operatorname{Set}^{\mathcal{C}^{\operatorname{op}}}$ to $\operatorname{Set}^{\mathcal{D}^{\operatorname{op}}}$. Over here, there is a strictly associative composition, so life is good.

All of this extends easily to the enriched setting. In particular, if we enrich over $\mathcal{V} = \operatorname{Cat}$, we can talk about strict 2-profunctors between strict 2-categories, and we get a nice bicategory of 2-categories, strict 2-profunctors, and morphisms between these.

## Question

I would like to know if there is an analogous way to obtain a tricategory of bicategories and biprofunctors. You can "follow your nose" and write down what the composition should be, but checking all the axioms of a tricategory does not seem like the simplest approach. Of course, we can strictify to get equivalent strict 2-categories and 2-profunctors, but I doubt the composition will be the same; I don't think the "strict" colimits will in general agree with the "weak" colimits. Also, using the standard enriched theory only gives a bicategory; the natural transformations are lost. It seems that an appropriate statement of the form "the bicategorical Yoneda embedding is a free cocompletion" will handle things just as it does in the setting of ordinary profunctors, but I am not sufficiently comfortable with limits and colimits in bicategories to try to figure out if such a statement makes sense. Has anybody worked all of this out?

(The closest thing I have seen to such a result is this paper by Justin Greenough, which more or less establishes this result in the case of sufficiently nice $k$-linear monoidal categories. But his proof seems to be very specific to that setting, whereas one would hope that a proof could be established along the same lines as the result for usual profunctors.)