There is an abstract construction of the coskeleton functor for simplicial set as follows:
Fix an integer $n\geq 0$ and take the inclusion of the full subcategory $i_n\colon\Delta_{\leq n}\hookrightarrow \Delta$. The induced functor $$ i_{n*}\colon sSet\to sSet_{\leq n} $$ given by $X\mapsto X\circ i_n$ has a right-adjoint $i_n^!\colon sSet_{\leq n}\to sSet$ and the composition $\mathbf{coskel}_n=i_n^!\circ i_{n*}$ is the $n$-coskeleton functor and it comes with a natural map $X\to \mathbf{coskel}_n(X)$.
I want to do the same for the pointed situation: Let $n\geq 0$ be an integer and cosinder the functor $$ i'_{n*}\colon sSet_*\to sSet_{*\leq n} $$ given by $X\mapsto X\circ i_n$ for a pointed simplicial set $X\in sSet_*=*\downarrow sSet\cong\operatorname{Fun}(\Delta^{op},Set_*)$. This has a right-adjoint ${i_n'}^{!}\colon sSet_{*\leq n}\to sSet_*$. Let the composition $\mathbf{coskel'}_n={i_n'}^{!}\circ i_{n*}'$ be the pointed $n$-coskeleton functor. It comes with a natural map $X\to \mathbf{coskel}'_n(X)$.
Is $X\to \mathbf{coskel}'_n(X)$ for a pointed simplicial set $X\in sSet_*$ an isomorphism on the simplicial homotopy group $\pi_i(-):=[\Delta^i/\partial\Delta^i, -]_{pointed}$ for $0\leq i<n$ and is $\pi_i(\mathbf{coskel}'_n(X))=0$ for $i\geq n$?
For the unpointed situation this seem to be true as stated here where an isomorphism of the simplicial homotopy groups mean an isomorphism with respect to all basepoints. The question is thus probably easy to answer but I am unsure if there is a picky basepoint-detail going wrong in the pointed case. Thank you.