Hallo,
I have to worry you one more time with these acyclicity problems, but as I am currently working on derived functors in a.g., I really need to understand derived functors in a very general form.
So my question is just: I have an exact functor $F: K^{+}(A)\rightarrow K(B)$ of the homotopy categories of abelian categories A and B and I know that the derived $RF: D^{+}(A)\rightarrow D(B)$ exists (in the sense of: it is exact and has universal property). Then this does not a priori imply that for each complex in $K^{+}(A)$ I can find a quasiiso to a complex of F-acyclics? I myself would guess that one has to have a triangulated subcategory L of $K^{+}(A)$ which is adapted to F, in the sense of Hartshorne, Residues and Duality. A short hint to if I am right is totally enough, thanks.