For a chain complex \(C_*\) in an Abelian category with enough injectives, and a left exact functor T, one has the classical Künneth map \(\phi_ n: H_ nTC_*\to TH_ nC_*\). By imposing the strong restriction that \(T^{(i)}(Z_ nC\to H_ nC)\) is a split monomorphism for every i, and assuming as usual that \(C_ n\) is T-acyclic for all n, the author writes down a long exact sequence which incorporates \(\phi_ n\), and whose other terms are of the form \(T^{(j)}H_{n+j}\). The proof is elementary and straightforward. Applications are indicated to singular and Massey cohomology theories of topological spaces.