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I have added the version of the axioms for reduced cohomology here at generalized (Eilenberg-Steenrod) cohomology (and also at reduced cohomology), and I have further expanded and streamlined (I hope) the Idea-section.
More harmonization (notation, conventions) is necessary in this and related entries. Later.
I have been editing generalized (Eilenberg-Steenrod) cohomology further:
expanded and streamlined the definitions;
added various basic lemmas, with proofs;
added the construction of reduced from unreduced cohomology and vice versa
spelled out the core of the proof that this gives an equivalence
(didn’t write out proof that all maps are compatible, as in the big diagrams of Switzer’s lemma 7.42, 7.44)
have also added in the statement of the Mayer-Vietoris long exact sequence for unreduced cohomology here. For the moment without proof.
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