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.