Kerodon

Definition 8.3.2.9. 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 from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$. We say that $\mathscr {K}$ is representable if, for each object $Y \in \operatorname{\mathcal{C}}_{+}$, the functor $\mathscr {K}(-, Y): \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is representable (in the sense of Variant 5.6.6.2). We will say that $\mathscr {K}$ is corepresentable if, for each object $X \in \operatorname{\mathcal{C}}_{-}$, the functor $\mathscr {K}(X, -): \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ is corepresentable (in the sense of Definition 5.6.6.1).