Proposition 8.3.4.15. Let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a functor of $\infty $-categories, where $\operatorname{\mathcal{C}}$ is locally small, and let $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ be a profunctor. The following conditions are equivalent:
- $(1)$
The profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$ is representable by $G$, in the sense of Definition 8.3.4.2.
- $(2)$
There exists a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}( \operatorname{\mathcal{D}}) }$ which exhibits $\mathscr {K}$ as represented by $G$, in the sense of Definition 8.3.4.12.