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For differential manifold category, we can introduce the differential form to make up a cochain, and then get the de Rham cohomology group.

My question is that if we use $\text{Hom}$ functor to get the dual chain of the cochain, then does the new homology satisfies the Eilenberg-Steenrod axioms? If so, is there any reference to study it?

Any advice is helpful. Thank you.

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