A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories

Abstract: Given an extriangulated category $(\mathcal{C},\mathbb{E},\mathfrak{s})$, we introduce the $3 \times 3$-lemma property for subfunctors of $\mathbb{E}$ and prove that an additive subfunctor $\mathbb{F}$ of $\mathbb{E}$ 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 $\xi$ in $\mathcal{C}$.