Let $C$, $C'$, $D$, $D'$ be chain complexes, $f$, $f'\colon C\to C'$ and $g$, $g'\colon D \to D'$ two pairs of homotopic chain maps. How to show $f \otimes g$ and $f' \otimes g' \colon C\otimes D\to C'\otimes D'$ are homotopic? I have tried many formulas for such a homotopy, but none of them worked.
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