Kerodon

Corollary 4.6.9.20. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category, and let $U: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }$ be the isomorphism of homotopy categories supplied by Proposition 2.4.6.9. Then the homotopy equivalences $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }(X,Y)$ of Remark 4.6.8.6 promote $U$ to an isomorphism of $\mathrm{h} \mathit{\operatorname{Kan}}$-enriched categories. Here $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ is endowed with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of Remark 3.1.5.12 and $\mathrm{h} \mathit{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) }$ is endowed with the $\mathrm{h} \mathit{\operatorname{Kan}}$-enrichment of Construction 4.6.9.13.