  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.
