Given a left exact functor F between abelian categories A \to B, with a left adjoint G, what are some conditions to put on A, B and F which ensures that the map D(A) \to D(B) has a derived left adjoint? (We consider the unbounded derived category)

If I can take G-acyclic resolutions of unbounded complexes, we would be done. But the unbounded situation is not as clear; I think that adding exact filtered colimits in B helps, but I would rather want a criterion involving generators of A and B or something like that.

If D(A) is compactly generated it might be enough for D(F) to preserve coproducts. What if D(A) is generated by a proper class? Does the same still hold true?

Or some version of it