If $f \colon X \to Y$ is a proper monomorphism of schemes [monomorphism in the sense of category theory], then the underlying map of topological spaces is continuous, closed, and injective, hence a homeomorphism onto its image. Properness also implies finite type, which eliminates some of the more egregious monomorphisms of schemes. But I don't see any obvious argument (or counterexample) for why the corresponding morphism of sheaves $\mathcal{O}_Y \to f_* \mathcal{O}_X$ should be surjective.
Motivation: $X \to Y$ is a monomorphism iff the corresponding morphism of functors $\mathrm{Hom}(-,X) \to \mathrm{Hom}(-,Y)$ is injective. Thus, a positive answer to my question here would, in nice situations, allow for a valuative criterion as to whether a subfunctor is closed.
