Kerodon

Example 8.3.2.13. 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. If $\operatorname{\mathcal{C}}_{-} = \Delta ^0$, then the profunctor $\mathscr {K}$ is corepresentable (in the sense of Definition 8.3.2.9) if and only if it is corepresentable when regarded as a functor $\operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ (in the sense of Definition 5.6.6.1). Similarly, if $\operatorname{\mathcal{C}}_{+} = \Delta ^0$, then the profunctor $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9) if and only if it is representable when viewed as a functor $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ (in the sense of Variant 5.6.6.2).