Kerodon

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Proposition 4.8.8.14. 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 $G: \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \rightarrow \operatorname{\mathcal{D}}$ of Construction 4.8.8.10 is an $n$-categorical inner fibration (see Definition 4.8.6.23).

Proof. For $n < 0$, this is immediate from the construction. We may therefore assume without loss of generality that $n \geq 0$. Using Remarks 4.8.6.32 and 4.8.8.13, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In particular, $\operatorname{\mathcal{D}}$ is an $(n,1)$-category. In this case, Example 4.8.8.11 guarantees that the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})} \simeq \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{D}})}$ is an $(n,1)$-category. The desired result now follows from Proposition 4.8.6.31. $\square$