Internal Hom of module sheaves

Let $X$ be a scheme and let $\mathcal{F},\mathcal{G}$ be ${\mathcal{O}}_X$-modules. The internal Hom of $\mathcal{F}$ and $\mathcal{G}$ is the sheaf $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$ of The internal Hom sheaf of two module sheaves, whose sections over an open $U\subseteq X$ are $\mathrm{\Gamma }(U,\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G}))={\mathrm{Hom}}_{{\mathcal{O}}_U}(\mathcal{F}{\mathrm{\mid }}_U,\mathcal{G}{\mathrm{\mid }}_U),$ the ${\mathcal{O}}_X(U)$-module of ${\mathcal{O}}_U$-linear maps, with restrictions given by restriction of morphisms. It is a sheaf of ${\mathcal{O}}_X$-modules, and its formation is contravariant in $\mathcal{F}$ and covariant in $\mathcal{G}$.

This definition introduces no quasi-coherence claim. When $\mathcal{F}$ is finitely presented and quasi-coherent (Finite type and finitely presented module sheaves, Quasi-coherent module on a scheme) and $\mathcal{G}$ is quasi-coherent, the lemma on this page proves that $\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$ is again quasi-coherent; for an arbitrary quasi-coherent $\mathcal{F}$ no such claim is made here, and none is used on this page. The functor $\mathcal{G}\mapsto \mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$ is left exact for every $\mathcal{F}$, because the functor $V\mapsto \mathrm{Hom}$ is left exact, and the global sections are ${\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$; the stalk at a point $x$ is not $\mathrm{Hom}({\mathcal{F}}_x,{\mathcal{G}}_x)$ in general.