Kerodon

Remark 8.3.2.11 (Symmetry). 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 $\operatorname{\mathcal{C}}_{-}$. Then $\mathscr {K}$ is representable if and only if it is corepresentable when regarded as a profunctor from $\operatorname{\mathcal{C}}_{-}^{\operatorname{op}}$ to $\operatorname{\mathcal{C}}_{+}^{\operatorname{op}}$ (see Remark 8.3.2.3).