Remark 8.3.2.16. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}$ be a covariant transport representation for $\lambda $. Using Remark 8.3.2.8, we deduce the following:
The coupling $\lambda $ is representable (in the sense of Definition 8.2.1.3) if and only if the profunctor $\mathscr {K}$ is representable (in the sense of Definition 8.3.2.9).
The coupling $\lambda $ is corepresentable (in the sense of Definition 8.2.1.3) if and only if the profunctor $\mathscr {K}$ is corepresentable (in the sense of Definition 8.3.2.9).