**Question.** Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of $\mathcal{O}_X$-modules. Is there always a quasicoherent sheaf $\mathcal{E}'$ together with a monomorphism $\mathcal{E} \to \mathcal{E}'$?

**Remark.** The coherator yields a way to find a quasicoherent sheaf together with a morphism *to* $\mathcal{E}$. But I'm interested in finding a quasicoherent sheaf together with a monomorphism *from* $\mathcal{E}$.

**Motivation.** There is a way to set up the theory of sheaf cohomology for quasicoherent sheaves without injective or flabby resolutions. If any sheaf of modules would embed into a quasicoherent one, we might be able to extend this development to arbitrary (not necessarily quasicoherent) sheaves of modules.