Sorry if this has already been answered somewhere, but I can't find it if it has.
Let $(X,\mathcal{O}_X)$ be a ringed space, and $\mathcal{F},\mathcal{G}$ two sheaves of $\mathcal{O}_X$-modules.
Is there some nice way of simplifying $$\mathcal{H}\mathrm{om}(\mathcal{F},\mathcal{G})\otimes \mathcal{H}\mathrm{om}(\mathcal{F},\mathcal{G})?$$
If needed, we can assume that $\mathcal{F}$ is locally free, and possibly (but ideally not) that $\mathcal{G}$ is locally free.