Example 4.8.8.11. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of $\infty $-categories and let $n$ be an integer. Then there is a comparison map from the simplicial set $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to the homotopy $n$-category $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. For $n \geq 0$, this map carries an $m$-simplex $(\sigma ,\tau )$ of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}$ to the $m$-simplex of $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ given by the composite map
If $\operatorname{\mathcal{D}}$ is an $(n,1)$-category, then this comparison map is an isomorphism (Proposition 4.8.4.20).