An elementary proof of a criterion for subfunctors of Ext to be closed

Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the additive bifunctor ${\text{Ext}}_{\mathcal{A}}^1(-,-)\text{\hspace{0.17em}⁣}:{\mathcal{A}}^{\text{op}}\times \mathcal{A}\to \mathsf{A}\mathsf{b}$. Buan proved in [4] that $F$ is closed if, and only if, $F$ has the $3\times 3$-lemma property, a certain diagrammatic property satisfied by the class of $F$-exact sequences. The proof of this result relies on the theory of exact categories and on the Freyd–Mitchell embedding theorem, a very well-known overpowered result. In this paper we provide a proof of Buan’s result only by means of elementary methods in abelian categories. To achieve this we survey the required theory of subfunctors leading us to a self-contained exposition of this topic.