Loop space has all abelian homotopy groups $i \geq 1$, done via simplicial sets

Question

This is a proof from Goerss-Jardine (p.31):

What do they mean by the multiplication $\star$? Should I think of $\pi_n(\Omega X, *)$ as $1$ simplices in $\mathsf{sSet}(\Delta^n, X)$? In order to do my homotopy stuff, I need a Kan complex. Now I'm working with the subset of these which give the constant map at the identity when I precompose with any coface map. Should I try to show this subset is Kan somehow (akin to how $\Omega X$ is Kan?) Some diagram sketching didn't get me there.

Answer 1

As user Kevin Carlson pointed out to me, the space we're interested in is Kan because it is a fiber of the fibration $\mathsf{sSet}(\Delta^n, X) \to \mathsf{sSet}(\partial \Delta^n, X)$.