Proposition 3.5.4.22. Let $X$ be a simplicial set. For every integer $n$, the tautological map $q: \operatorname{cosk}_{n+1}(X) \rightarrow \operatorname{cosk}_{n}^{\circ }(X)$ is a trivial Kan fibration.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix an integer $m \geq 0$; we wish to show that every lifting problem
3.73
\begin{equation} \begin{gathered}\label{equation:strong-coskeleton-vs} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{cosk}_{n+1}(X) \ar [d]^{q} \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{cosk}_{n}^{\circ }(X) } \end{gathered} \end{equation}
admits a solution. We consider two cases:
If $m \leq n+1$, then $\overline{\sigma }$ can be lifted to an $m$-simplex of $X$. In particular, there exists an $m$-simplex $\sigma $ of $\operatorname{cosk}_{n+1}(X)$ satisfying $q(\sigma ) = \overline{\sigma }$. Since $q$ is bijective on $k$-simplices for $k \leq n$, the commutativity of the diagram (3.73) guarantees that $\sigma |_{ \operatorname{\partial \Delta }^{m} } = \sigma _0$.
If $m \geq n+2$, then $\sigma _0$ extends uniquely to an $m$-simplex $\sigma $ of $\operatorname{cosk}_{n+1}(X)$. The commutativity of diagram (3.73) guarantees that $q(\sigma )$ and $\overline{\sigma }$ have the same restriction to $\operatorname{\partial \Delta }^{m}$, and therefore coincide (since $\operatorname{\partial \Delta }^ m$ contains the $(n+1)$-skeleton of $\Delta ^ m$).
In either case, the $m$-simplex $\sigma $ is a solution to the lifting problem (3.73). $\square$