$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 4.8.2.6. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. For every integer $n$, the following conditions are equivalent:
The homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is locally $n$-truncated.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is $n$-truncated.
Proof.
For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, Theorem 4.6.8.5 supplies a homotopy equivalence from $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ to the pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}( X, Y)$. The desired result now follows from Remark 4.8.2.5.
$\square$