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.



How does a chain map induce another chain map on an isomorphic chain complex?
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.


