Natural isomorphism for a reduced cohomology theory on CW complexes. I've found here: http://www.people.fas.harvard.edu/~xiyin/Site/Notes_files/AT.pdf the following lemma.
$\textbf{Lemma 2.}$ If a reduced cohomology theory $h^{*}$ defined on C complexes has $h^{n}(S^{0})=0$ for $n \neq 0$, then there are natural isomorphisms
\begin{align*}
h^{n}(X) \cong \overline{H^{n}}(X;h^{0}(S^{0}))
\end{align*}
for all CW complexes X and integer $n$.
Where I can find a proof of this lemma or how to prove it?
Please for help with this problem.
 A: In the linked paper it is not defined which axioms a reduced cohomology theory should satisfy. Although reduced axiomatic (co)homology is a fairly standardized concept, I would say that there is only basic axiom (the exactness axiom) and a variety of optional axioms. Examples for such axioms are the dimension axiom and the wedge axiom. See e.g. Chapter 7 of
Switzer, Robert M. Algebraic topology-homotopy and homology. Springer, 2017.
If $h^n(S^0) = 0$ for $n \ne 0$, then $h^*$ satisfies the dimension axiom and is called an ordinary cohomology theory with coefficients in $G = h^0(S^0)$. It is well-known that any two ordinary (co)homology theory with the same coefficient group $G$ are naturally isomorphic on finite CW-complexes. If in addition the wedge axiom is satisfied, then the theories are naturally isomorphic on all CW-complexes. See Switzer Chapter 10. Note that in Switzer most results are stated for homology, but the cohomology versions are also true (because proofs transfer). Also note that the wedge axiom is only needed for infinite wedges (for finite wedges you can prove it from the other axioms).
