nLab
cograph of a profunctor

Let $H:A⇸B$ be a profunctor, i.e. a functor $B^{\mathrm{op}}\times A\to \mathrm{Set}$. Its cograph, also called its collage, is the category $\overline{H}$ whose set of objects is the disjoint union of the sets of objects of $A$ and $B$, and where

$$\begin{array}{rl}\overline{H}(a_1,a_2)& =A(a_1,a_2)\\ \overline{H}(b_1,b_2)& =B(b_1,b_2)\\ \overline{H}(b,a)& =H(b,a)\\ \overline{H}(a,b)& =\varnothing \end{array}$$\begin{aligned} \bar{H}(a_1,a_2) &= A(a_1,a_2)\\ \bar{H}(b_1,b_2) &= B(b_1,b_2)\\ \bar{H}(b,a) &= H(b,a)\\ \bar{H}(a,b) &= \emptyset \end{aligned}

where composition is defined as in $A$, $B$, and according to the actions of $A$ and $B$ on $H$.

The cograph of a functor is the special case when $H$ is a “representable profunctor” of the form $B(f-,-)$ for some functor $f:A\to B$.

The cograph of a profunctor can be given a universal property: it is the lax colimit of that profunctor, considered as a single arrow in the bicategory of profunctors. (The word “collage” is also sometimes used more generally for any lax colimit, especially in a $\mathrm{Prof}$-like bicategory.) The cograph of a profunctor is also a cotabulation? in the proarrow equipment of profunctors. Furthermore, the cospans $A\to \overline{H}\leftarrow B$ which are cographs of profunctors can be characterized as the two-sided codiscrete cofibrations in the 2-category Cat.

All of this can be generalized to enriched categories and also to higher categories.