Corollary 4.8.4.16. Let $n \geq 0$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-faithful functor of $\infty $-categories. Then, for every diagram $B \rightarrow \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{Fun}_{/\operatorname{\mathcal{D}}}(B, \operatorname{\mathcal{C}})$ is locally $(n-2)$-truncated.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. It follows from Corollary 4.1.4.2 that $F$ induces an inner fibration $F': \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ and from Corollary 4.8.4.15 that $F'$ is $n$-faithful. Applying Proposition 4.8.3.36, we deduce that every fiber of $F'$ is locally $(n-2)$-truncated. $\square$