For any ascending chain $\Omega$ of subobjects of a fixed object $M$, the supremum of $\Omega$ exists; and, for any subobject $L\hookrightarrow M$, the canonical morphism

$sup\{L\cap P | P\in \Omega\}\to (sup \Omega) \cap L$

According to the Appendix B of the Thomason–Trobaugh contribution to Grothendieck Festschrift, it is not known whether the category $Qcoh_Y$ of quasicoherent sheaves over an arbitrary scheme $Y$ 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).