Proposition 4.8.4.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be a positive integer. Then the comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ of Construction 4.8.4.9 exhibits $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Corollary 4.8.4.14, it will suffice to show that for every $(n,1)$-category $\operatorname{\mathcal{D}}$, the composite map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}, \operatorname{\mathcal{D}}) \xrightarrow {\theta } \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \xrightarrow {\theta '} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \]
is a bijection. Since $\operatorname{\mathcal{D}}$ is weakly $n$-coskeletal, the map $\theta '$ is a bijection. By construction, $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$ is a quotient of the weak coskeleton $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$, so $\theta $ is an injection. We will complete the proof by showing that $\theta $ is also a surjection: that is, every diagram $F: \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ factors through $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. Let $\sigma $ and $\sigma '$ be $m$-simplices of $\operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}})$ satisfying $\sigma \sim _{m} \sigma '$ (see Construction 4.8.4.9); we wish to show that $F(\sigma ) = F(\sigma ')$. This follows from the observation that $\operatorname{\mathcal{D}}$ is minimal in dimension $n$ (Proposition 4.8.1.7). $\square$