Construction 3.5.6.10. Let $n$ be a nonnegative integer, let $X$ be a Kan complex. and let $\operatorname{cosk}^{\circ }_{n}(X)$ denote the weak $n$-coskeleton of $X$ (Notation 3.5.4.19). For every integer $m \geq 0$, we will identify $m$-simplices of $\operatorname{cosk}^{\circ }_ n(X)$ with morphisms $\sigma : \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow X$ which can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$ (see Remark 3.5.4.21). Given two such morphisms $\sigma , \sigma ': \operatorname{sk}_{n}( \Delta ^ m ) \rightarrow X$, we write $\sigma \sim _{m} \sigma '$ if $\sigma $ and $\sigma '$ are homotopic relative to $\operatorname{sk}_{n-1}( \Delta ^{m} )$. The construction
determines a simplicial set, which we will denote by $\pi _{\leq n}(X)$. By construction, we have an epimorphism of simplicial sets $q: \operatorname{cosk}_{n}^{\circ }(X) \twoheadrightarrow \pi _{\leq n}(X)$, which determines a comparison map $X \rightarrow \pi _{\leq n}(X)$.