Corollary 4.8.6.23. Let $n \geq -1$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an essentially $n$-categorical inner fibration 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 $(n-1)$-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 4.8.6.22 that $F'$ is essentially $n$-categorical. In particular, every fiber of $F$ is locally $(n-1)$-truncated. $\square$