A characterization of closed subfunctors through (3× 3)-lemma property in extriangulated categories

Given an extriangulated category (C,E,s) ( C , E , s ) , we introduce the (3 × 3) ( 3 × 3 ) -lemma property for subfunctors of E E and prove that an additive subfunctor F F of E E is closed if, and only if, it satisfies this condition. This characterization extends a well known result by A. Buan (for abelian categories) to extriangulated categories. As an application of this result, we get a new equivalent condition to describe saturated proper classes ξ ξ in C C .