Kerodon

Remark 8.3.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A profunctor $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.3.1) if and only if there exists a natural transformation $\alpha : \underline{ \Delta ^0 }_{\operatorname{Tw}(\operatorname{\mathcal{C}}) } \rightarrow \mathscr {H}|_{ \operatorname{Tw}(\operatorname{\mathcal{C}})}$ which exhibits $\mathscr {H}$ as a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$ (in the sense of Definition 8.3.5.1).