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

Remark Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. The preceding proof shows that if $\operatorname{\mathcal{C}}$ satisfies the conclusion of Proposition, then it also satisfies the conclusion of Theorem 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 (applied to the simplicial category $\operatorname{Kan}$) implicitly to give the definition of a corepresentable functor in the $\infty $-categorical setting.