Why bar resolution is acyclic complex of $G$-homomorphisms?

Question

I have a little confusion in showing the bar resolution of $\mathbb{Z}$ (as $G$-module) acyclic. One constructs the complex $X_{\star}$ of free $G$-module, where $X_r$ is free $G$-module generated by the symbols $[g_1,\cdots,g_r]$, $g_i\in G$. This can be viewed as free $\mathbb{Z}$-module over $g[g_1,\cdots,g_r]$. Then one construct the differentials $\partial_r:X_{r}\rightarrow X_{r-1}$. In show that it acyclic complex of $G$-homomorhism, one construct the homotopies as $\mathbb{Z}$-homomorphism and shows identity map is null homotopy. But this construction will only show that is acyclic complex of $\mathbb{Z}$-homomorphisms not as $G$-homomorphisms. Why is it also acyclic complex of $G$-homomorphisms?