Intermediate categories for proper abelian subcategories

Let $\mathcal{A}$ be an extension closed proper abelian subcategory of a triangulated category $\mathcal{T}$, with no negative 1 and 2 extensions. From this, two functors from $\mathrm{\Sigma }\mathcal{A}\ast \mathcal{A}$ to $\mathcal{A}$ can be constructed giving a snake lemma mirroring that of homology without needing a t-structure. We generalize the concept of intermediate categories, which originates from a paper by Enomoto and Saito, to the setting of proper abelian subcategories and show that under certain assumptions this collection is in bijection with torsion-free classes in $\mathcal{A}$.