Construction 4.8.4.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $n$ be a positive integer, and let $\operatorname{cosk}^{\circ }_{n}(\operatorname{\mathcal{C}})$ denote the weak $n$-coskeleton of $\operatorname{\mathcal{C}}$ (Notation 3.5.4.19). For every integer $m \geq 0$, we will identify $m$-simplices of $\operatorname{cosk}^{\circ }_ n(\operatorname{\mathcal{C}})$ with diagrams $\sigma : \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow \operatorname{\mathcal{C}}$ which can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$ (Remark 3.5.4.21). Given two such morphisms $\sigma , \sigma ': \operatorname{sk}_{n}( \Delta ^ m ) \rightarrow \operatorname{\mathcal{C}}$, we write $\sigma \sim _{m} \sigma '$ if $\sigma $ and $\sigma '$ are isomorphic relative to $\operatorname{sk}_{n-1}( \Delta ^{m} )$ (Definition 4.7.6.1). The construction
determines a simplicial set, which we will denote by $\mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$. By construction, it is equipped with an epimorphism of simplicial sets $ \operatorname{cosk}_{n}^{\circ }(\operatorname{\mathcal{C}}) \twoheadrightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$, which determines a comparison map $\operatorname{\mathcal{C}}\rightarrow \mathrm{h}_{\mathit{\leq n}}\mathit{(\operatorname{\mathcal{C}})}$.