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I have 2 ways of defining a chain complex on a manifold, one of which is the cellular chain complex, $C^{CW}_*$. I know that a cellular map $f: X^n \rightarrow Y^n $ such that $ f(X^n) \subset Y^n $ for all $ n $ induces a map on the cellular chain complexes $ f_* : C^{CW}_* (X^n) \rightarrow C_*^{CW}(Y^n) $ through naturality of $ f $ with respect to the two exact sequences which form the boundary homomorphism of $ C^{CW}_* $.

I also have an isomorphism between the two chain complexes: $I: C_*^{CW} \rightarrow D_* $ which commutes with the boundary operators: $ I \circ \partial^{CW}_* $ = $ \partial^D_* \circ I $.

Since the chain complexes are isomorphic, I assume that $f$ will also induce a chain map $f_D$ on $D_*$, but I'm not sure how to show this formally.

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up vote 1 down vote accepted

Assuming $I\colon C^{CW}\to D$ is a natural isomorphism between functors from CW-complexes to chain complexes, just set

$$f_D = I\circ f_*\circ I^{-1}.$$

This should work.

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