Dg algebras with enough idempotents, their $\mathrm{dg}$ modules and their derived categories

We develop the theory $\mathrm{dg}$ algebras with enough idempotents and their $\mathrm{dg}$ modules and show their equivalence with that of small $\mathrm{dg}$ categories and their $\mathrm{dg}$ modules. We introduce the concept of $\mathrm{dg}$ adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as $\mathrm{dg}$ adjunctions between categories of $\mathrm{dg}$ 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 $\mathrm{dg}$ algebra with enough idempotents, the perfect left and right derived categories are dual to each other.