Here is the problem:
Suppose $R$ is a field.
(a) Show that $h^{n}(?) = Hom_{R}(H_{n}(?; R), R)$ is a cohomology theory defined on (at least) the category of finite CW complexes.
(b) Show that $u$ is the natural transformation of cohomology theories.
Where $u$ is defined in the following paragraph:
And we also should have solved the following problem before trying to solve 22.39(b):
Show that is natural in both variables. That is suppose $f: X \rightarrow Y, u \in \tilde{H^{*}}(Y), \alpha \in \tilde{H_{*}}(X).$ Then we can form $$<u, f_{*}(\alpha)> \in \tilde{H}_{n-k}(Y)$$ And $$<f^{*}(u), \alpha)> \in \tilde{H}_{n-k}(X).$$
Where $<?,?>$ is the Pairing Cohomology with homology.
Could anyone tell me how may 22.35 help me in the solution of (b)?
Also, could anyone help me in the solution in general?