Weakly equivalence preserving functors

Let R be a commutative ring and T be an additive covariant functor from the category of R-modules to itself. We recall that an R-module M is said to be T-acyclic (resp. T-coacyclic) if the n-th right (resp. left) derived functor of T on M is zero for any positive integer n. Assume that T is a left (resp. right) exact functor and X, Y are two bounded to the left (resp. right) R-complexes of T-acyclic (resp. T-coacyclic) R-modules such that X Y. We will show that T(X) T(Y). As an application, we extend the main results of Sazeedeh, Divaani-Aazar and Mohammadi that provide some new ways for computing local (co)homology modules.