Given a fibrant simplicial set $X$ (has lifting condition with all horns) and a vertex $v \in X_0$.
I want to show that the simplicial homotopy group $\pi_n(X,v)$ satisfies the inverse axiom. The composition law for two map $f,g:\Delta^n \rightarrow X$ is given in nlab.
I could show the identity axiom. Assuming associativity, it suffices to prove
The left multiplication map $$ \pi_n(X,v) \rightarrow \pi_n(X,v), \quad [x] \mapsto [f][x] $$ is bijective.
This is stated in Goerss, simplicial homotopy theory. How does one show bijectivity?