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 [XR] the ideal of morphisms factoring through XR=Ker(HomC(R,−)), then applications of our results show that C/[XR] 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.
