Internal Hom sheaf ($\mathcal{H}o{m}_{{\mathcal{O}}_X}(\mathcal{F},\mathcal{G})$)

For sheaves of modules $\mathcal{F},\mathcal{G}$ on a ringed space, the internal Hom sheaf assigns to an open $U$ the module of sheaf morphisms $\mathcal{F}{|}_U\to \mathcal{G}{|}_U$ respecting ${\mathcal{O}}_U$. Compatible morphisms glue, so this is a sheaf. Its global sections are precisely the global module-sheaf morphisms. For a locally free sheaf $\mathcal{F}$ of finite rank it is ${\mathcal{F}}^{\vee }\otimes \mathcal{G}$.