Let $M$ be a smooth manifold and $E,F,G$ smooth vector bundles over $M$. Denote the global sections by $\Gamma(.)$.
In this question, it is proven that the canonical map $$\Gamma(E)\otimes_{C^\infty(M)}\Gamma(F)\rightarrow\Gamma(E\otimes F)$$ is an isomorphism of $C^\infty(M)$ modules.
There is also the canonical morphism of $C^\infty(M)$ modules given by $$\Gamma(G^*)\rightarrow Hom_{C^\infty (M)}(\Gamma(G),C^\infty(M)),\omega\mapsto[X\mapsto \omega(X)]$$
Is this map an isomorphism, too? If not, is it an iso under certain conditions, for example if $M$ is compact?