  Let $\mathcal{A}$ be an abelian category and let $F$ be a subbifunctor of the
additive bifunctor $\text{Ext}_{\mathcal{A}}^{1}(-,-)\colon
\mathcal{A}^{\text{op}}\times \mathcal{A}\to \mathsf{Ab}$. 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.
