Problem 22.39(c) in "Modern Classical Homotopy Theory " by Jeffery Strom on pg.511.

Question

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.

(c) Prove Theorem 22.37.

And here is the theorem:

Could anyone help me in proving so, please?