Corollary 4.8.1.20. Let $n$ be an integer and let $\operatorname{\mathcal{C}}$ be an $(n,1)$-category. Then, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For $n \leq -1$, there is nothing to prove (see Examples 4.8.1.10 and 4.8.1.9). The case $n = 0$ follows from Proposition 4.8.1.15. We may therefore assume $n > 0$. By virtue of Proposition 4.6.5.10, it will suffice to show that the pinched morphism space $K = \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{C}}}(X,Y)$ is $(n-1)$-truncated. This follows from Example 3.5.7.2, since $K$ is an $(n-1)$-groupoid (Proposition 4.8.1.19). $\square$