Does the functor $\mathbf{cosk_n}:sSet\to sSet$ preserve Kan complexes? On this nlab page a functor $\mathbf{cosk_n}:sSet\to sSet$ is constructed.

Is $\mathbf{cosk_n}(K)$ of a Kan complex $K$ again a Kan complex?

 A: $\mathbf{cosk}_n$ does indeed preserve Kan complexes.
Let $\mathbf{sk}_n$ be the left adjoint of $\mathbf{cosk}_n$. Given a horn inclusion $\Lambda^m_k \hookrightarrow \Delta^m$, the morphism $\mathbf{sk}_n (\Lambda^m_k \hookrightarrow \Delta^m)$ is an isomorphism if $n > m + 1$, so by adjointness, $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^m_k \hookrightarrow \Delta^m$ for all $n > m + 1$, whether or not $K$ is a Kan complex. 
On the other hand, $\mathbf{sk}_n (\Lambda^m_k \hookrightarrow \Delta^m)$ is (canonically isomorphic to) $\Lambda^m_k \hookrightarrow \Delta^m$ itself when $m \le n$, so $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^m_k \hookrightarrow \Delta^m$  if and only if $K$ does, when $m \le n$. 
That leaves $m = n + 1$. Here, $\mathbf{sk}_n (\Lambda^{n+1}_k \hookrightarrow \Delta^{n+1})$ is isomorphic to $\Lambda^{n+1}_k \hookrightarrow \partial \Delta^{n+1}$, so $\mathbf{cosk}_n (K)$ has the right lifting property with respect to $\Lambda^{n+1}_k \hookrightarrow \Delta^{n+1}$ if and only if $K$ has the right lifting property with respect to $\Lambda^{n+1}_k \hookrightarrow \partial \Delta^{n+1}$. This is not an anodyne extension, but Kan complexes nonetheless have the right lifting property with respect to it: indeed, choose a horn filler as usual and restrict along the inclusion $\partial \Delta^{n+1} \hookrightarrow \Delta^{n+1}$.
