Let $X = $ Spec $A$ be an affine scheme. I know that there is an inclusion of categories from $A$-modules to sheaves of $\mathcal O_X$-modules on $X$, which is exact and fully faithful.
It seems to me that by Lemma 17.10.5./(4) of the stacks project (https://stacks.math.columbia.edu/tag/01BH), this inclusion functor has a right adjoint (the global section functor), hence $A$-modules form a reflective subcategory. The image of the inclusion functor is precisely the category of quasi-coherent sheaves on $X$.
My question is: does this generalize to arbitrary (non-affine) schemes? This would immediately give a nice proof that quasi-coherent sheaves on any scheme form an Abelian category.
The category of quasicoherent sheaves is actually a coreflective subcategory of the category of all sheaves of modules on the scheme.
See the stacks project.