Kerodon

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

Theorem 4.6.7.5. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Then the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 4.6.7.3 is a homotopy equivalence of Kan complexes.