Suppose I have a cospan of categories $X\xrightarrow{F} Z\xleftarrow{G}Y$ and I'm looking for exact squares containing it. The category $Exact\_{F,G}$ of such things has a terminal object, namely the comma category $(F\downarrow G)$. In fact, the comma category construction provides a section of the forgetful functor $U\colon Exact\to Cspn$ from the category of exact squares to the category of cospans.


But what else is known about this category $Exact\_{F,G}$? I would like to know about completeness, cocompleteness, etc. 


Are there cases (i.e. subcategories of special cospans $F,G$) for which something interesting can be said about $Exact$? For example, if we look at cospans for which $F$ is a discrete opfibration, there is another section of $U$, given by 1-categorical pullback. 


My motivation comes from the case when all categories in sight are finite. Given a cospan, my real interest is in finding exact squares for which the upper-left-hand category is as small as possible.



Exact squares containing a cospan: what is known about this category?
Not a complete answer, but a few observations: $\newcommand{\Cat}{\mathbf{Cat}} \newcommand{\dn}{\downarrow} \newcommand{\Ex}{\mathrm{Exact}}$


* since it’s a full subcategory of the category of lax-commutative squares over $(F,G)$, it’s isomorphic to a full subcategory of $\Cat / (F \dn G)$. Also, $\Cat / (F \dn G)$ is easily seen to be complete and cocomplete, since $\Cat$ is.
* this subcategory is of the form “all objects which, under each of a class of functors $R\_M : \Cat / (F \dn G) \to [M^Y, M^X]$, become isomorphic to the terminal object $1\_{(F \dn G)}$”, ~~where each $R\_M$ is a right adjoint. It is therefore closed under limits in $(F \dn G)$, so it’s complete.~~ **Edit:** this and the next point were mistaken; the precompose-then-Kan-extend functors are continuous (resp. co-continuous below) in their $M^Y$ (resp. $M^X$) arguments, but not in the $\Cat / (F \dn G)$ argument.
* it’s also of the form “all objects which, under each of a class of functors $L\_M :\Cat / (F \dn G) \to [M^X, M^Y]$, become isomorphic to the terminal object”, ~~where each $L\_M$ is a left adjoint. So by the same argument, it’s co-complete.~~
* understanding $\mathrm{Exact}(F,G)$ in more detail reduces — again via factorisation through the comma category — to understanding it in the case where $F$ is an identity morphism, and (if desired) where we restrict to *strictly* commutative exact squares. Precisely, $\Ex(F,G) \cong \Ex\_{\mathit{str}}(1\_X,F^\*G) \simeq \Ex(1\_X,F^\*G)$, where $F^\*G$ is the projection from the comma category to $X$.
* one special case of this is (when $X \simeq 1$) the question of understanding the final subcategories of a given category, which I have a vague recollection of being told is a hard question in general — I’m not very sure of this memory, though.


