Kerodon

Remark 8.3.3.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let let $\mathscr {H}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for $\operatorname{\mathcal{C}}$. Passing to homotopy categories, we obtain a functor $H: \mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \times \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$. It follows from Corollary 8.1.2.18 (and Remark 5.6.5.10) that $H$ is isomorphic to the functor $(X,Y) \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ determined by the $\mathrm{h} \mathit{\operatorname{Kan}}$ enrichment of the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (see Construction 4.6.9.13). See Remark 8.3.5.4 for a more precise statement.