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Given a DG-category A we introduce the bar category of modules Modbar(A). Itis a DG-enhancement of the derived category D(A) of A which is isomorphic tothe category of DG A-modules with A-infinity morphisms between them. However,it is defined intrinsically in the language of DG-categories and requires nocomplex machinery or sign conventions of A-infinity categories. We define forthese bar categories Tensor and Hom bifunctors, dualisation functors, and aconvolution of twisted complexes. The intended application is to working withDG-bimodules as enhancements of exact functors between triangulated categories.As a demonstration we develop homotopy adjunction theory for tensor functorsbetween derived categories of DG-categories. It allows us to show in anenhanced setting that given a functor F with left and right adjoints L and Rthe functorial complex $FR \rightarrow FRFR \rightarrow FR \rightarrow Id$lifts to a canonical twisted complex whose convolution is the square of thespherical twist of F. We then write down four induced functorial Postnikovtowers computing this convolution.

Bar category of modules and homotopy adjunction for tensor functors