Let $(X, \mathcal{O}_X)$ ringed space, $\mathcal{F}, \mathcal{G}$ invertible sheaves on $X$ and $\underline{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) $ the $Hom$ - sheaf.
There is always given the canonical evaluation map $\mathcal{F} \otimes \underline{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) \to \mathcal{G}$.
My question is how to prove that this map is an isomorphism?
This is because we have a canonical isomorphism $Hom_{O_X}(F, G) \cong Hom_{O_X}(F,O_X) \otimes G$ for $F,G$ invertible sheaves on $X$ if $X$ is noetherian which I am assuming. (This holds when $X$ is noetherian, $F$ is locally free and $G$ coherent).
Under this isomorphism you want to check that the map $$ F \otimes Hom_{O_X}(F,O_X) \otimes G \to G $$
is an isomorphism. This map is the composite of $\text{ev}_F \otimes id_G : F \otimes Hom_{O_X}(F,O_X) \otimes G \to O_X \otimes G$, the evaluation map tensored with $id_G$, and the multplication map $O_X \otimes G \to G$. Both maps are isomorphisms so you get an isomorphism. I hope it helps.