General heart construction for twin torsion pairs on triangulated categories

In our previous article, we constructed an abelian category from any torsion pair on a triangulated category. This generalizes the heart of a $t$-structure and the ideal quotient by a cluster tilting subcategory. Recently, generalizing the quotient by a cluster tilting subcategory, Buan and Marsh showed that an integral preabelian category can be constructed as a quotient, from a rigid object in a triangulated category with some conditions. In this article, by considering a pair of torsion pairs, we make a simultaneous genralization of these two constructions.