Example 8.3.4.5. Let $\operatorname{\mathcal{C}}$ be a locally $\kappa $-small $\infty $-category and let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{<\kappa }$ be a profunctor from $\operatorname{\mathcal{C}}$ to itself. The following conditions are equivalent:
The profunctor $\mathscr {H}$ is a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$.
The profunctor $\mathscr {H}$ is representable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Definition 8.3.4.2).
The profunctor $\mathscr {H}$ is corepresentable by the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ (Variant 8.3.4.4).