Kerodon

Proposition 8.3.4.19. Suppose we are given a functor of $\infty $-categories $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$, a profunctor $\mathscr {K}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}$, and a natural transformation $\beta : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{D}})} \rightarrow \mathscr {K}|_{ \operatorname{Tw}(\operatorname{\mathcal{D}})}$. Then $\beta $ exhibits $\mathscr {K}$ as represented by $G$ (in the sense of Definition 8.3.4.12) if and only if the induced map $\operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \{ \Delta ^0 \} \operatorname{\vec{\times }}_{ \operatorname{\mathcal{S}}} ( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{D}})$ is left cofinal.