A "natural homomorphism" $H^n(B; \Bbb Z) \rightarrow H^n(B; \Bbb Z_2)$.

Question

This is Proposition 4.12, pg33 The claim of the statement is:

The natural homomorphism $\Gamma:H^n(B;\Bbb Z) \rightarrow H^n(B ; \Bbb Z_2)$ sends the Euler class to the top Stiefel Whitney class.

My confusion, is, what exactly is this natural homomorphism?

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Is this given by taking a cochain $\varphi:C_n(B) \rightarrow \Bbb Z$ and post compose by quotient map - if so, is it clear this well defined?

Answer 1

We have a natural transformation $Hom(-,\mathbb Z) \to Hom(-,\mathbb Z/2)$ of endofunctors of abelian groups induced by the quotient map. Applying this to the chain complex $C_n(B;\mathbb Z)$ determines the morphism of cochain complexes $$C^n(B; \mathbb Z) \to C^n(B; \mathbb Z/2)$$ morphisms of complexes always induce morphisms on cohomology.