Question about Neukirch's book Cohomology of Number Fields

Question

Let $G$ be a profinite group. $A$ and $B$ are discrete G-modules. If $A^U=A$ for some open subgroup $U \subseteq G,$ then why Hom(A,B) is a discrete G-module. $\big( g(\phi)(a) = g(\phi(g^{-1}(a))) \big)$