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

Let A be an abelian category and let F be a subbifunctor of the additive bifunctor ExtA¹(-,-) Aᵒᵖ× A→ Ab. Buan proved in [4] that F is closed if, and only if, F has the 3× 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.