Given a space $X$, and an Eilenberg-MacLane space $K(G,n)$ (hereafter referred to as $K$), and two maps $f: X \to K$ and $g:X \to K$, let $f \times g:X \to K \times K$ map $x \in X$ to $(f(x),g(x))$. Is it the case that, in the induced map on $H^n(-,G)$, $(f \times g)^* (\alpha) = f^*(\alpha) + g^*(\alpha)$ for a generator $\alpha$ of $H^n(K \times K;G)$, where $f^*$ is the map on cohomology induced by $f \times 0$, and $g^*$ is the map on cohomology induced by $0 \times g$?
It certainly seems that it should be case, but I'm having trouble proving it. I know that for an Eilenberg-MacLane space $K(G,n)$, the universal coefficient theorem and Hurewicz theorem show that $H^n(K(G,n);G) = Hom(H_n(K(G,n)),G) = Hom(\pi_n(K(G,n),G) = Hom(G,G)$ (and, by a similar token, $H^n(K(G,n) \times K(G,n);G) = Hom(G \times G,G)$, but I don't know where to go from here. I'd greatly appreciate any help.