Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same true for $f \otimes g : C \otimes D \to C' \otimes D'$?
The answer is yes if $C$ and $C'$ consist of flat modules, because then we may use the Künneth spectral sequence and its functoriality. I wonder what happens in the general case?