The T-equivalence and J-iterations introduced for a bifunctor Tenable one to deduce a "shifting property" from which one gets the "long exact sequence of homology." Also, the appropriate lemma of Schanuel is proved, from which one can develop a T-dimension theory. These notions are useful in proving known duality homomorphisms and may serve to get some new ones. For one purpose, they unify the methods of studying the various common dimensions.