Proposition 4.8.6.35. Let $n$ be an integer and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-categorical inner fibration of $\infty $-categories. Then $F$ is essentially $n$-categorical.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 4.8.6.35. If $n = -2$, then $F$ is an isomorphism of simplicial sets and therefore an equivalence of $\infty $-categories (Example 4.5.1.11). If $n = -1$, then $F$ is an isomorphism from $\operatorname{\mathcal{C}}$ onto a full subcategory of $\operatorname{\mathcal{D}}$, and therefore fully faithful (Example 4.6.2.2). We may therefore assume without loss of generality that $n \geq 0$. By virtue of Proposition 4.8.6.17, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a standard simplex; in this case, we wish to show that $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. This follows from Example 4.8.2.2, since $\operatorname{\mathcal{C}}$ is an $(n,1)$-category (Proposition 4.8.6.32). $\square$