Let $R$ be a commutative ring. For a fixed CW complex $Y$, consider the functors $$ h^n(X)=\oplus _{i=0} ^n (H^i(X;R) \otimes_R H^{n-i} (Y;R) ) ,$$ $$k^n(X)=H^n (X \times Y;R).$$
(Here $X$ is a CW complex.)
In the proof of Proposition 3.17 in Hatcher's Algebraic Topology, Hatcher says that when $n$ is sufficiently large with respect to $i$, then the maps $h^i(X) \to h^i(X^n)$ and $k^i(X) \to k^i(X^n)$ induced by the inclusion $X^n \to X$ are isomorphisms.
I cannot see why does this result holds. But I think that the similar results from homology may imply this result, via the universal coefficient theorem for cohomology. I know that the inclusion $X^n \to X$ induces an isomorphism on homology $H_k(X^n) \to H_k(X)$ is $k<n$. Can I prove the result using this? Or is there another way to prove the result?