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 $\mathcal{ C}$.
Categories: math.CT math.RT