TITLE: Adjunctions between derived functors
QUESTION [6 upvotes]: 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 $LF\dashv RG$?

REPLY [3 votes]: 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