In the Lecture Notes in Algebraic Topology by Davis & Kirk, we consider the Kronecker pairing which is defined by
$$H^n(C;R) \times H_n(C;R) \to R \\ ([\varphi],[\alpha]) \mapsto \varphi(\alpha)$$
where $H^n(C;R)$ is the cohomology and $H_n(C;R)$ is the homology with coefficients over a ring $R$.
We know this defines a map $$H^n(C;R) \to \operatorname{Hom}(H_n(C;R),R) \\ [\varphi] \mapsto ([\alpha] \mapsto \varphi(\alpha)). $$
Davis & Kirk want to prove that cohomology is not the dual of homology in general and mentioned that The map $H^n(C;R) \to \operatorname{Hom}(H_n(C;R),R)$ does not need to be injective nor surjective.
My question: Why is it enough to consider this specific map?
If cohomology is the dual of homology, then $H^n(C;R) \simeq \operatorname{Hom}(H_n(C;R),R)$. So why can't there be another bijective map $H^n(C;R) \to \operatorname{Hom}(H_n(C;R),R)$ then?
Thank you in advance!