Kerodon

Remark 8.3.4.6. Let $\lambda : \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+}$ be a coupling of $\infty $-categories which is essentially $\kappa $-small for some uncountable cardinal $\kappa $ and let $\mathscr {K}: \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a covariant transport representation for $\lambda $. Then the profunctor $\mathscr {K}$ is representable by a functor $G: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}_{-}$ (in the sense of Definition 8.3.4.2) if and only if the coupling $\lambda $ is representable by $G$ (in the sense of Definition 8.2.3.1). Similarly, $\mathscr {K}$ is corepresentable by a functor $F: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}_{+}$ if and only if $\lambda $ is corepresentable by $F$.