Corollary 3.5.4.23. Let $X$ be a simplicial set and let $n$ be an integer. If $X$ is a Kan complex, then the weak $n$-coskeleton $\operatorname{cosk}_{n}^{\circ }(X)$ is a Kan complex.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Proposition 3.5.4.22 supplies a trivial Kan fibration
\[ q: \operatorname{cosk}_{n+1}(X) \twoheadrightarrow \operatorname{cosk}_{n}^{\circ }(X). \]
Since $\operatorname{cosk}_{n+1}(X)$ is a Kan complex (Proposition 3.5.3.23), it follows that $\operatorname{cosk}_{n}^{\circ }(X)$ is also a Kan complex (Proposition 1.5.5.11). $\square$