  We develop the theory dg algebras with enough idempotents and their dg
modules and show their equivalence with that of small dg categories and their
dg modules. We introduce the concept of dg adjunction and show that the
classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules
over associative unital algebras are extended as dg adjunctions between
categories of dg bimodules. The corresponding adjunctions of the associated
triangulated functors are studied, and we investigate when they are one-sided
parts of bifunctors which are triangulated on both variables. We finally show
that, for a dg algebra with enough idempotents, the perfect left and right
derived categories are dual to each other.
