Proposition 4.8.6.17. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets and let $n$ be an integer. Then the comparison map $G: \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \rightarrow \operatorname{\mathcal{D}}$ of Construction 4.8.6.13 is an $n$-categorical inner fibration (see Definition 4.8.6.1).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. For $n < 0$, this is immediate from the construction. We may therefore assume without loss of generality that $n \geq 0$. Using Remarks 4.8.6.10 and 4.8.6.16, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In particular, $\operatorname{\mathcal{D}}$ is an $n$-category. In this case, Example 4.8.6.14 guarantees that the simplicial set $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \simeq \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}})}}$ is an $n$-category. The desired result now follows from Proposition 4.8.6.9. $\square$