Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext.-1(X, Y ) = 0 for each object X . X and each object Y . Y. We show that if A,B . C .(R) are X-complexes and U, V . C .(R) are Y-complexes, then . As an application, we give a sufficient condition for the Hom evaluation morphism being invertible.