Global sections of the Hom sheaf

Question

Let $F$ and $G$ be $O_X$-modules, and consider the Hom sheaf $\mathcal{H}om_{O_X}(F,G)$. What are the global sections of this sheaf?

I know that there is an isomorphism of $O_X(X)$-modules $$\Gamma(X,\mathcal{H}om_{O_X}(F,G)) \cong \mathrm{Hom}_{O_X-mod}(O_X,\mathcal{H}om_{O_X}(F,G)).$$

Is there any way to view the right hand side as $\mathrm{Hom}_{O_X-mod}(F,G)$?

Answer 1

By definition, the module of global sections of the $\mathcal{Hom}(\mathcal{F},\mathcal{G})$ is the $\mathcal{O}(X)$-module of morphisms of sheaves $\varphi: \mathcal{F}\rightarrow \mathcal{G}$.