  We prove that, under a mild assumption, the heart H of a twin cotorsion pair
((S,T),(U,V)) on a triangulated category C is a quasi-abelian category. If C is
also Krull-Schmidt and T=U, we show that the heart of the cotorsion pair (S,T)
is equivalent to the Gabriel-Zisman localisation of H at the class of its
regular morphisms.
  In particular, suppose C is a cluster category with a rigid object R and
[X_R] the ideal of morphisms factoring through X_R=Ker(Hom(R,-)), then
applications of our results show that C/[X_R] is a quasi-abelian category. We
also obtain a new proof of an equivalence between the localisation of this
category at its class of regular morphisms and a certain subfactor category of
C.
