Let $A$ and $B$ be abelian categories and let $F : A \rightarrow B$ be a left exact functor. We can form the right derived functors $R^i F$ of $F$ by taking an injective resolution $X \rightarrow I_0 \rightarrow I_1 \rightarrow \cdots$, applying $F$, and taking cohomology. Call an object $X$ in $A$ $F$-flat if $R^i F(X)$ vanishes for $i >0$. There's probably a real name for these objects, but I don't know it.
Here is my question: can we take a resolution of $F$-flat objects to calculate $R^i F$? Also, are resolutions via $F$-flat objects unique up to homotopy equivalence?
This is true for example with flasque sheaves. Flasque sheaves are the sheaves whose cohomology vanishes. When calculating cohomology, we can take a resolution via flasque sheaves instead of injective ones. For a sheaf $F$, we have a flasque sheaf $F^\#$ defined where $F^\#(U) = \prod_{x \in U} F_x$. We have a monad whose unit is $F \rightarrow F^\#$ and that monad gives us a Bar resolution in $\text{Ch}(A)$. This example motivates the question of whether we can do this in a more general setting. Here it allowed for a canonical resolution in $\text{Ch}(A)$.