where composition is defined as in , , and according to the actions of and on .
The cograph of a functor is the special case when is a “representable profunctor” of the form for some functor .
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 -like bicategory.) The cograph of a profunctor is also a cotabulation? in the proarrow equipment of profunctors. Furthermore, the cospans which are cographs of profunctors can be characterized as the two-sided codiscrete cofibrations in the 2-category Cat.
The notion of a cograph of a profunctor generalizes to (∞,1)-category theory.
This appears as (Lurie, def 188.8.131.52).
There is a canonical bijection between equivalence classes of correspondences between and and equivalence classes of -profunctors (∞,1)-functors
This appears as (Lurie, remark 184.108.40.206).
Therefore the correspondence corresponding to a profunctor is its cograph/collage.
Because by the (∞,1)-Grothendieck construction
and as discussed at adjoint (∞,1)-functor, an -functor has an adjoint precisely if the coCartesian fibration corresponding to it is also Cartesian.
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 -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.