An abelian category has property (sup) if:
is an isomorphism.
Gabriel’s property (sup) is satisfied by any Grothendieck category (in some expositions it is listed as a part of the definition), e.g. the category of all modules over a fixed ring , and the category of sheaves of abelian groups on a fixed topological space .
According to the Appendix B of the Thomason–Trobaugh contribution to Grothendieck Festschrift, it is not known whether the category of quasicoherent sheaves over an arbitrary scheme is a Grothendieck category (although this is known for a large class), but it is an elementary fact that it does satisfy Gabriel’s property (sup). If an abelian category is noetherian it clearly satisfies the property (sup).