By a (co)homology theory on the category of (finite) CW pairs I mean a functor from the category of (finite) CW pairs to the category of graded $R$-modules, satisfying the Eilenberg-Steenrod (co)homology axioms (including the dimension axiom). The existence part of this is done in any standard algebraic topology textbook by showing that the singular (co)homology is a actually a (co)homology theory.
I want to have some reference for the uniqueness part. To be precise, I want to prove the following (and similar results for cohomology theories) :
$H_\star$ and $\hat{H}_\star$ be two homology theories, $p$ be a point. Then for any homomorphism $H_0(p)\to \hat{H}_0(p)$, there is a natural transformation $H_\star\to\hat{H}_\star$ inducing the given map. If $H_0(p)\to\hat{H}_0(p)$ is given to be an isomorphism, the natural transformation induces isomorphism $H_\star(X,A)\to\hat{H}_\star(X,A)$ for any pair $(X,A)$ i.e $H_\star$ and $\hat{H}_\star$ becomes equivalent.
Can someone please provide me some reference for this? Any help is appreciated.
This is not true. For example, complex K-theory has the same value at a point as 2-periodic integral cohomology, but they are not isomorphic as cohomology theories. I believe there isn't even a map of cohomology theories between them which gives an isomorphism on a point. (There is such a map after rationalization, which is given by the Chern character. But after rationalization both theories become 2-periodic rational cohomology.)
The correct statement (I think; I'm not 100% on this) is only that evaluation at a point is conservative: that is, given a natural transformation of cohomology theories, if it's an isomorphism on a point then it's an isomorphism. This is a version of the Whitehead theorem, but for spectra (although it might only be true for connective spectra...).