Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 4.6.6.6. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Combining Theorem 4.6.6.5 with Proposition 4.6.5.9, 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 4.6.6.3 with the left-pinch inclusion map of Construction 4.6.5.6.