I see that most authors prove only the Künneth formula for Homology, and get the formula for cohomology as a consequence. My question here is how to get from one to the other. I assume you need to use the Universal Coefficient Theorem. So My approach is the following.
For all that follows assume $X$ and $Y$ are topological spaces, $R$ is a PID and $M$ is an $R$-module.
The two things I understand is that in this conditions we have the following short exact split squences
The Universal Coefficient Theorem for Cohomology $$0\rightarrow \text{Ext}_R(H_{n-1}(X;R),M) \rightarrow H^n(X;M) \rightarrow \text{Hom}_R(H_n(X;R),M) \rightarrow 0$$
The Künneth formula for Homology $$0 \rightarrow \bigoplus_{i=0}^n H_i(X;R)\otimes H_{n-i}(Y;R) \rightarrow H_n(X\times Y) \rightarrow \bigoplus_{i=0}^{n-1} \text{Tor}(H_i(X;R), H_{n-1-i}(Y;R)) \rightarrow 0$$
From what I have seen the cohomology version is pretty much the same exact sequence, just replacing $H_k$ with $H^k$ everywhere in the sequence above.
My question is, how do you use what I have mentioned to arrive at the cohomology version of the Künneth formula?