The notion of cograph of a functor is dual to that of graph of a functor: for $f : C \to D$ a functor between n-categories is the fibration classified by the profunctor correspondence $\chi_f : C^{op}\times D \to (n-1)Cat$. But $f$ also determines a morphism $\bar f : I \to n Cat$ from the interval category $I$. The cograph of $f$ is the fibration classified by $\bar f$.
Recall that the graph of a function $f: A \to B$ is the subset determined by the monomorphism $\langle 1, f \rangle: A \to A \times B$. This makes sense in any category with products. Under one definition, the notion of cograph of a function $f: A \to B$ is the categorically dual notion: it is the quotient determined by the epimorphism $(f, 1): A \sqcup B \to B$.
A more vivid presentation of cograph is given by the pictures we draw of functions $f: A \to B$: as directed graphs (in the graph-theoretic sense!) whose vertices are elements in $A \sqcup B$, with an edge drawn from $a$ to $f(a)$ for each $a$ in $A$. This can also be conceived as a poset $P_f$ with underlying set $A \sqcup B$, in which $a \leq f(a)$ and all other instances of $\leq$ are the reflexive ones. The connected components function $(P_f)_0 \to \pi_0(P_f)$ is then the cograph in the sense given above.
In this article we give a definition of cograph which generalizes this poset picture of cograph of a function, and which applies to any functor between $n$-categories.
In the case that $C, D$ are 0-categories, i.e. sets, a functor $f : C \to D$ is just a function between sets. The cograph 2-pullback
is computed by the ordinary pullback
and identifies $cograph(f)$ with the category of elements of $\bar f$, as described there: the objects of $cograph(f)$ are the disjoint union of $C$ and $D$: $Obj(cograph(f)) = C \coprod D$ and the nontrivial morphisms are of the form $x \to y$ whenever $x \in C$, $y \in D$ and $f(x) = y$.
What Bill Lawvere called the cograph of a function is the connected components $\pi_0(Cograph(f))$ of this category.
For $f : C \to D$ an ordinary functor, $cograph(f)$ is the category with $Obj(cograph(f)) = Obj(C) \coprod Obj(D)$ and with
with composition defined as induced from $C$, from $D$, and from the action of $f$. This is a special case of the cograph of a profunctor, specialized to the representable profunctor $D(f-,-)$.
This cograph is denoted $C \star^f D$ in section 2.3.1 (_Correspondences_) in
where it is understood as a generalization of the join of simplicial sets and where it serves as a motivation for the study of cographs of functors between (∞,1)-categories discussed below.
As emphasized in the beginning of section 5.2 there, cographs of functors may be used to characterize adjoint functors. This is just one way of stating the characterization of adjoints in terms of profunctors (which in turn makes sense in any 2-category equipped with proarrows).
Two functors $L : C \to D$ and $R : D \to C$ are adjoint functors precisely if $cograph(L)$ and $cograph(R^{op})^{op}$ are isomorphic under $C$ and $D$:
where the isomorphism is in the co-slice category $(C\sqcup D)/Cat$.
More precisely, there is a bijection between adjunctions $L\dashv R$ and isomorphisms as above.
The category $cograph(L)$ is the category with $Obj(cograph(L)) = Obj(C) \coprod Obj(D)$ and with
(This set is also called the set of heteromorphisms between objects in $C$ and $D$.)
The category $cograph(R^{op})^{op}$ accordingly is the category with $Obj(cograph(R)) = Obj(C) \coprod Obj(D)$ and with
Evidently these categories are isomorphic under $C$ and $D$ precisely if for all $x \in C, y \in D$ we have
naturally in $x$ and $y$. It is natural because the isomorphism is an isomorphism of categories, and the functoriality of $Hom_D(L(-),-)$ and $Hom_C(-,R(-))$ is encoded by composition in the cograph. Of course, such a natural isomorphism is precisely the structure of an adjunction $L\dashv R$.
Note also that just as $cograph(L)$ is the cograph of the profunctor $D(L-,-)$, also $cograph(R^{op})^{op}$ is the cograph of the profunctor $C(-,R-)$. Thus, this theorem can be viewed as one way of stating the characterization of adjunctions in terms of homsets, as can be formulated in terms of profunctors in any 2-category equipped with proarrows.
In the context of (∞,1)-category theory there is a good theory of Cartesian fibrations $X \to S$ and of their classification by (∞,1)-functors $S^{op} \to (\infty,1)Cat$ to the (∞,1)-category of (∞,1)-categories as described at universal fibration of (∞,1)-categories.
Accordingly, the above notion of cograph of a functor has a direct generalization to (∞,1)-functors:
For $f : C \to D$ an (∞,1)-functor, identified with a morphism
in the (∞,1)-category of (∞,1)-categories, it cograph is the Cartesian fibration $cograph(f) \to I$ classified by it. In terms of the universal fibration of (∞,1)-categories this is the homotopy pullback
As for every Cartesian fibration the functor $f : C \to D$ is determined uniquely up to equivalence by its cograph. In general, obtaining the classifying $(\infty,1)$-functor from a given Cartesian fibration may be difficult. In the special case of cographs as Cartesian fibrations over the simple interval category it is easier. This is discussed in the following:
…
cograph of a functor, cograph of a profunctor
The notion of cographs of $(\infty,1)$-functors and the theory of how to re-obtain $(\infty,1)$-functors from their cographs is the content of section 5.2.1, Correspondences and associated functors, of