Universal coefficient theorem in triangulated categories

Let T be a triangulated category, A a graded abelian category and h: T -> A a homology theory on T with values in A. If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category and the ideal Ker(h) is a square zero ideal which as a bifunctor on T is isomorphic to Ext^1_A(h(-)[1], h(-)).