Remark 4.6.8.6. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Combining Theorem 4.6.8.5 with Proposition 4.6.5.10, we obtain a homotopy equivalence of Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$, given by composing the comparison map $\theta $ of Construction 4.6.8.3 with the left-pinch inclusion map of Construction 4.6.5.7.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$