ABSTRACT: We prove two triangle equivalences. One is the triangle equivalence between the homotopy category of the bounded below complexes of Ext-injectives objects of a closed by subobjects and co-resolving subcategory of an abelian category and the derived category of the bounded below complexes over The other triangle equivalence is between the homotopy category of the bounded cohomology and bounded below complexes over a strongly closed by cokernels of monomorphisms and auto-orthogonal subcategory of an abelian category and the derived category of the bounded cohomology and bounded below complexes overCOL0086896COL0017217