Why is $\operatorname{Fun}(A^{*},\mathbb C^{*})$ injective?

Question

Everything below is a $\mathbb Z-$mod.

Let $A^{*}$ = $(\operatorname{Hom}A,\mathbb C^{*})$.
I(A) = $\operatorname{Fun}(A^{*},\mathbb C^{*})$. [note : only functions,not homomorphisms]

Why is I(A) an injective module?

I know that $\mathbb C^{*}$ is injective (as it is divisible).