Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, respectively below) derived categories. What can one say about the existence of an adjunction $LF\dashv RG$?
It seems this question was considered in:
William W Adams, Marc A Rieffel. Adjoint functors and derived functors with an application to the cohomology of semigroups Journal of Algebra, V. 7, N 1, 1967, 25-34