Proposition 3.5.4.26. Let $X$ be a Kan complex. Then, for every integer $n$, the tautological map $u: X \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof of Proposition 3.5.4.26. Fix a pair of integers $0 \leq i \leq m$ with $m > 0$; we wish to show that every lifting problem
3.74
\begin{equation} \begin{gathered}\label{equation:fibration-to-strong-coskeleton} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{ \sigma _0 } \ar [d] & X \ar [d]^{u} \\ \Delta ^{m} \ar@ {-->}[ur]^{\sigma } \ar [r]^-{ \overline{\sigma } } & \operatorname{cosk}^{\circ }_{n}(X) } \end{gathered} \end{equation}
admits a solution. We consider two cases:
If $m \leq n+1$, then we can choose an $m$-simplex $\sigma $ of $X$ satisfying $u(\sigma ) = \overline{\sigma }$. Since $u$ is bijective on simplices of dimension $\leq n$, the commutativity of the diagram (3.74) guarantees that $\sigma |_{ \Lambda ^{m}_{i} } = \sigma _0$.
If $m \geq n+2$, then our assumption that $X$ is a Kan complex guarantees that $\sigma _0$ can be extended to an $m$-simplex $\sigma $ of $X$. The commutativity of the diagram (3.74) then guarantees that $u(\sigma )$ and $\overline{\sigma }$ have the same restriction to the horn $\Lambda ^{m}_{i} \subset \Delta ^{m}$. In particular, they have the same restriction to the $n$-skeleton of $\Delta ^{m}$, so $u(\sigma ) = \overline{\sigma }$.
In either case, it follows that $\sigma $ is a solution to the lifting problem (3.74). $\square$