Kerodon

Remark 8.3.2.12. Let $\operatorname{\mathcal{C}}_{-}$ and $\operatorname{\mathcal{C}}_{+}$ be $\infty $-categories and let $\mathscr {K}$ and $\mathscr {K}'$ be profunctors from $\operatorname{\mathcal{C}}_{+}$ to $\operatorname{\mathcal{C}}_{-}$ which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}, \operatorname{\mathcal{S}})$). Then $\mathscr {K}$ is representable if and only if $\mathscr {K}'$ is representable. Similarly, $\mathscr {K}$ is corepresentable if and only if $\mathscr {K}'$ is corepresentable. See Remark 5.6.6.4.