  We use the framework of Tensor-Hom-Cotensor situations to present a
generalization to abelian categories of the flat cotorsion theory for modules
over a ring. Using the generalized flat cotorsion theory we present a flat
totally acyclic cotorsion theory in categories of chain complexes in abelian
categories. Under some assumption on the generalized flat cotorsion theory we
prove a K\"unneth type theorem for chain complexes in abelian categories
