I [learned](https://mathoverflow.net/questions/252698/does-bf-prof-admit-all-pseudolimits) that the bicategory $\bf Prof$, while having lots of interesting properties, does not admit inverters, so it does not admit arbitrary pseudolimits.


Does this imply that it is impossible to define some useful constructions, like comma objects?


More in general, is there a precise account of which co/limits, bi-co/limits, and weighted co/limits can be constructed in $\bf Prof$?


In fact, in the thread above I say that $\bf Prof$ admits $\bf Cat$-cotensors, and I'm quite sure that ${\cal A}\pitchfork \bf B$ is defined on objects by $({\cal A},{\bf B})\mapsto {\bf A}^°\otimes {\bf B}$ (boldface = 1-cells of $\bf Prof$; mathcal = 1-cells of $\bf Cat$); but I'm not able to see what how to build the canonical span $d\_0,d\_1 \colon {\cal I}\pitchfork {\bf B} \rightsquigarrow \bf B$ in the case ${\cal I}=\{0\to 1\}$: how is it possible to define two functors ${\cal B}\otimes{\cal I}^°\otimes {\cal B}^° \to {\cal V}$ that satisfy a similar property of evaluation on co/domain?



Comma objects in the bicategory of profunctors
I don't know about comma objects, and I don't think there is a known characterization of which limits and colimits exist in Prof.


But cotensors do exist and are given as you say. More generally, lax limits of lax functors exist and coincide with lax colimits, the projections being the right adjoints of the coprojections. In particular, $d\_0,d\_1 : \mathcal{I} \pitchfork \mathbf{B} \nrightarrow \mathbf{B}$ are the corepresentable profunctors adjoint to the representable profunctors induced by the coprojections $i\_0,i\_1 : \mathbf{B} \to \mathbf{I}^{\mathrm{op}} \times \mathbf{B}$.


See for instance Street's paper *Cauchy characterization of enriched categories* and later work such as Carboni-Kasangian-Walters *An axiomatics for bicategories of modules*. Relatedly, [this paper](https://arxiv.org/abs/1301.3191) of my own with Garner exhibits Prof as a certain kind of free cocompletion.


