Kerodon

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

Remark 5.6.6.19. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. The preceding proof shows that if $\operatorname{\mathcal{C}}$ satisfies the conclusion of Proposition 5.6.6.18, then it also satisfies the conclusion of Theorem 4.6.7.5: that is, the comparison map $\theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y)$ is a homotopy equivalence for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$. Note however that we have already used Theorem 4.6.7.5 (applied to the simplicial category $\operatorname{Kan}$) implicitly to give the definition of a corepresentable functor in the $\infty $-categorical setting.