Let $H\colon A ⇸B$ be a profunctor, i.e. a functor $B^{op}\times A\to Set$. Its cograph, also called its collage, is the category $\bar{H}$ whose set of objects is the disjoint union of the sets of objects of $A$ and $B$, and where
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 $f\colon A\to B$.
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 $Prof$-like bicategory.) The cograph of a profunctor is also a cotabulation? in the proarrow equipment of profunctors. Furthermore, the cospans $A\to \bar{H} \leftarrow B$ which are cographs of profunctors can be characterized as the two-sided codiscrete cofibrations in the 2-category Cat.
Cographs of profunctors can also be characterized as categories equipped with a functor to the interval category $(0\to 1)$, where $B$ is the fiber over $0$ and $A$ is the fiber over $1$. See Distributors and barrels.
The notion of a cograph of a profunctor generalizes to (∞,1)-category theory.
For $C$ and $D$ two (∞,1)-categories an correspondence between them is an $(\infty,1)$-category $p : K \to \Delta[1]$ over the interval category $\Delta[1] = \{0 \to 1\}$ with an equivalences $K_0 \simeq C$ and $K_1 \simeq D$.
This appears as (Lurie, def 2.3.1.3).
There is a canonical bijection between equivalence classes of correspondences between $C$ and $D$ and equivalence classes of $(\infty,1)$-profunctors (∞,1)-functors
from the product of $D$ with the opposite-(∞,1)-category of $C$ to ∞Grpd.
This appears as (Lurie, remark 2.3.1.4).
Therefore the correspondence corresponding to a profunctor is its cograph/collage.
An $(\infty,1)$-profunctor comes from an ordinary (∞,1)-functor $F : C \to D$ precisely if its cograph $p : K \to \Delta[1]$ is not just an inner fibration but a coCartesian fibration.
And it comes from a functor $G : D \to C$ precisely if it is a Cartesian fibration. And precisely if both is the case is $F$ the right adjoint (∞,1)-functor to $G$.
Because by the (∞,1)-Grothendieck construction
coCartesian fibrations $K \to \Delta[1]$ correspond to (∞,1)-functors $\Delta[1] \to$ (∞,1)Cat;
Cartesian fibrations $K \to \Delta[1]$ correspond to (∞,1)-functors $\Delta[1]^{op} \to$ (∞,1)Cat.
and as discussed at adjoint (∞,1)-functor, an $(\infty,1)$-functor has an adjoint precisely if the coCartesian fibration corresponding to it is also Cartesian.
cograph of a functor, cograph of a profunctor
For ordinary and enriched categories, cographs were studied (and used to characterize profunctors) by:
Ross Street, “Fibrations in bicategories”
Carboni and Johnson and Street and Verity, “Modulated bicategories”
The $(\infty,1)$-category theoretic notion (“correspondence”) is the topic of section 2.3.1 of
See Ross Street’s post in category-list 2009, Re: pasting along an adjunction.