Kerodon

Variant 8.3.4.16 (Corepresentable Profunctors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. We say that a natural transformation $\beta : \underline{ \Delta ^0 }_{ \operatorname{Tw}( \operatorname{\mathcal{C}}) } \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{C}})}$ exhibits $\mathscr {K}$ as corepresented by $F$ if, for every object $X \in \operatorname{\mathcal{C}}$, the image $\beta ( \operatorname{id}_{X} ) \in \mathscr {K}( X, F(X) )$ exhibits the functor $\mathscr {K}( X, - ): \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ as corepresented by the object $F(X) \in \operatorname{\mathcal{D}}$, in the sense of Definition 5.6.6.1. Equivalently, $\beta $ exhibits $\mathscr {K}$ as corepresented by $F$ if it exhibits $\mathscr {K}$ as represented by the opposite functor $F^{\operatorname{op}}$, when regarded as a profunctor from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{D}}^{\operatorname{op}}$ (see Remark 8.3.2.3).