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

Given an extriangulated category (C,E,s), we introduce the 3 × 3-lemma property for subfunctors of E and prove that an additive subfunctor F of 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 in C.