$K(G,n) \otimes K(G',n) \to K(G \times G',n)$

Question

Let $G,G'$ be two abelian groups and $X$ a topological space, we know that there is a natural bijection $$H^n(X,G) \cong [X, K(G,n)]$$ where $K(G,n)$ is an Eilenberg-Maclane space. Therefore, from the product $$H^n(X,G) \times H^n(X,G') \to H^n(X,G \times G').$$ We get a natural map $\boxtimes: [X, K(G,n) \times K(G',n)] = [X, K(G,n)] \times [X,K(G',n)] \to [X, K(G\times G',n)]$. Taking $X = K(G,n) \times K(G',n)$ then $$\boxtimes(id_X): K(G,n) \times K(G',n) \to K(G \times G',n).$$ I am struggling at proving this map induces an isomorphism on homology groups, any suggestion is appreciated. This comes from a paper of Seminar Cartan on DGA-algebras and DGA-modules, which is available here.