Proposition 4.8.8.18. 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 $F': \operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ of Remark 4.8.8.15 is an inner fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Without loss of generality, we may assume that $n \geq 0$. Using Remarks 4.1.1.13 and 4.8.8.13, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, $F'$ identifies with the tautological map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ (Example 4.8.8.11), so the desired result follows from Corollary 4.8.4.16. $\square$