I'm trying to prove the universal coefficient theorem (UCT) for homology and cohomology using spectral sequences following Section 5.6 of Weibel's book Introduction to homological algebra and coming up with the following problem:
Question: Let $C_{\bullet} = (C_{n} \xrightarrow{d} C_{n-1})$ be a chain complex and $P$ be a projective $R$-module. If each object $C_n$ is a flat $R$-module, then does the following equality (or say isomorphism) holds: $$ H_n (C_{\bullet} \otimes_{R} P) = H_n (C_{\bullet}) \otimes_{R} P. $$
Attempt: We shall note that every projective $R$-module is flat, so universal coefficient theorem does the trick. BUT I'm proving UCT, so we shouldn't use UCT to show this. Then how to show this without UCT?
Similarly, I came up with the following coquestion for the cohomology:
Coquestion: Let $C_{\bullet} = (C_{n} \xrightarrow{d} C_{n-1})$ be a chain complex and $I$ be an injective $R$-module. If each object $C_n$ is a projective $R$-module, then does the following equality (or say isomorphism) holds: $$ H^n (\mathrm{Hom}_R(C_{\bullet},I)) = \mathrm{Hom}_R(H_n(C_{\bullet}),I). $$
Coattempt: In this case, we even lost the trick of UCT, as we do not have the parallel statement for "projective $\Rightarrow$ flat", and clearly injectivity does not imply projectivity in general. So I doubt that actually the isomorphism in Question and Coquestion hold even without condition on $P$ and $I$ accordingly. Then how to show these isomorphisms?
Thank you all for commenting and answering! :)