It's probably worth commenting that $\mathbf{Prof}(A, B) \simeq \mathbf{Prof}(B°, A°)$, so a similar formula is true, but you need to introduce opposites.

Certain colimits of this form, called quasi-coproducts were studied in Hu–Tholen's Quasi-coproducts and accessible categories with wide pullbacks.

For later readers, David's first link is to On Morita Contexts in Bicategories.

To spell this out in different terminology, what you are describing are horizontal morphisms between cotabulators in the pseudo-double category of profunctors (taking horizontal morphisms to be the profunctors). All natural transformations between cocontinuous functors are "cocontinuous" in the appropriate sense, so I don't see that this corresponds to the concept you are looking for. Did you unwind the data of such a profunctor? That might make it easier to find other references.

@DavidCarchedi: the algebras for the small-cocompletion pseudomonad on $\mathbf{CAT}$ are locally-small categories with small colimits.

I haven't checked all the details, but this looks very much like multi-coclosed structure, i.e. that the tensor product has a multi-left adjoint. See Definition 2.8 of How nice are free completions of categories?, for instance (though your example is dual).

@MaximeRamzi: this is (a special case of) the proof of the coherence theorem for bicategories via Yoneda. I don't think it is reasonable to consider these 2-categories "naturally occurring", because otherwise Lack's observation becomes trivial. Of course, the observation isn't well-defined in the first place, so determining what "naturally occurring" even means is subtle, but I think it's fair to say that we should not expect any general coherence theorem to produce examples.

Isn't $Psh$ equivalent to $\mathbf{Cat}$ under the requirement that functors preserve tiny objects?