Exercise 8.3.2.14. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be ordinary categories. Show that a profunctor
\[ \mathscr {K}: \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{-})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_{+}) \rightarrow \operatorname{\mathcal{S}} \]
is representable (in the sense of Definition 8.3.2.9) if and only if it it is isomorphic to the profunctor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( X, G( Y) )$, for some functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$. See Proposition 8.3.4.1 for a more general result.