Given categories $X$ and $Y$ and a strong functor


$$D:X^{op}\times Y\to Cat$$


we can of course build the oplax colimit


$$\mathrm{colim}^{oplax}\_{X^{op}\times Y}D$$


via the usual (covariant) grothendieck construction:


*Objects* are triples $(x,d,y)$ with $d\in D(x,y)$.


*Morphisms* $(x,d,y)\to (x',d',y')$ are triples $(f,\phi, g)$ with $f:x'\to x$, $g:y\to y'$ and $$\varphi: f^\*x\_\*g\to x'.$$


There is however another possible construction that corresponds better to the slogan "presheaves are distributors into the one-point category and copresheaves are distributors out of the one-point category":


*Objects* are the same triples as above.


*Morphisms* however are now triples $(f,\varphi, g)$ with $f:x\to x'$ (the direction changed!) and $g:y\to y'$ and


$$\varphi:x\_\*g \to f^\*x'.$$


Why does it correspond better to the slogan stated above? Taking distributors having at one side the one-point category then specialises to the usual grothendieck constructions yielding fibered and opfibered categories respectively.


**Question:** What can be said about the second construction? Is it some kind of colimit as well?


[Related question](https://mathoverflow.net/questions/128387/composition-of-cat-valued-distributors-compatible-with-grothendieck-constructio)



grothendieck construction for profunctors
[I guess that by $x\_\*g$ you mean $D(\mathit{id}, g)(x)$ and by $f^\*y$ you mean $D(f, \mathit{id})(y)$.]


Actually, your second construction is the usual Grothendieck construction for "$\mathbf{Cat}$-valued distributors" (BTW, this term may be misleading a bit, because in a $\mathbf{Cat}$-valued distributor $\mathbb{X}^{op} \times \mathbb{Y} \rightarrow \mathbf{Cat}$ 2-categories $\mathbb{X}$ and $\mathbb{Y}$ are not necessary degenerated). It may be universally characterised as a $(- \downarrow \mathbb{Y}) \times (\mathbb{X} \downarrow =)$-weighted colimit of $D \colon \mathbb{X}^{op}\times \mathbb{Y} \rightarrow \mathbf{Cat}$.


See: "Cosmoi of Internal Categories" by Ross Street, Transactions American Math. Soc. 258 (1980) 271-318; MR82a:18007.


