Kerodon

Remark 8.3.2.15. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a profunctor. Then $\mathscr {K}$ is representable if and only if, for every object $Y \in \operatorname{\mathcal{C}}_{+}$, there exists an object $X \in \operatorname{\mathcal{C}}_{-}$ and a universal vertex $\eta \in \mathscr {K}( X,Y)$. Similarly, $\mathscr {K}$ is corepresentable if and only if, for every object $X \in \operatorname{\mathcal{C}}_{-}$, there exists an object $Y \in \operatorname{\mathcal{C}}_{+}$ and a couniversal vertex $\eta \in \mathscr {K}(X,Y)$.