General heart construction on a triangulated category (II): Associated cohomological functor

In the preceding part (I) of this paper, we showed that for any torsion pair (i.e., $t$-structure without the shift-closedness) in a triangulated category, there is an associated abelian category, which we call the heart. Two extremal cases of torsion pairs are $t$-structures and cluster tilting subcategories. If the torsion pair comes from a $t$-structure, then its heart is nothing other than the heart of this $t$-structure. In this case, as is well known, by composing certain adjoint functors, we obtain a cohomological functor from the triangulated category to the heart. If the torsion pair comes from a cluster tilting subcategory, then its heart coincides with the quotient category of the triangulated category by this subcategory. In this case, the quotient functor becomes cohomological. In this paper, we unify these two constructions, to obtain a cohomological functor from the triangulated category, to the heart of any torsion pair.