Kerodon

Proof. The tautological map $X \rightarrow \operatorname{cosk}_{n+1}(X)$ is bijective on $m$-simplices for $m \leq n+1$, so $f$ is surjective on $m$-simplices for $m \leq n+1$. Moreover, for $m \leq n$, the composite map $X \xrightarrow {f} \operatorname{cosk}_{n}^{\circ }(X) \hookrightarrow \operatorname{cosk}_{n}(X)$ is bijective on $m$-simplices for $m \leq n$, so that $f$ is injective on $m$-simplices. To complete the proof, it will suffice to show that $\operatorname{cosk}_{n}^{\circ }(X)$ is weakly $n$-coskeletal. Fix an integer $m > n$ and a morphism $\sigma _0: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$. Since $\operatorname{cosk}_{n}(X)$ is $n$-coskeletal, the morphism $\sigma _0$ extends uniquely to an $m$-simplex $\sigma $ of $\operatorname{cosk}_{n}(X)$. To complete the proof, it will suffice to show that if $m \geq n+2$, then $\sigma $ is contained in $\operatorname{cosk}_{n}^{\circ }(X)$: that is, that it can be lifted to an $m$-simplex of $\operatorname{cosk}_{n+1}(X)$. Using Remark 3.5.3.21, we can identify $\sigma $ with a morphism of simplicial sets $u: \operatorname{sk}_{n}( \Delta ^{m} ) \rightarrow X$; we wish to show that $u$ can be extended to the $(n+1)$-skeleton of $\Delta ^{m}$. By virtue of Proposition 1.1.4.12, this is equivalent to the requirement that, for every nondegenerate $(n+1)$-simplex $\tau $ of $\Delta ^{m}$, the composite map

can be extended to an $(n+1)$-simplex of $X$. This follows from our assumption that $\sigma _0 \circ \tau $ factors through $\operatorname{cosk}_{n}^{\circ }(X)$. $\square$