Hey i'm looking for a different Version of the Grothendieck Spectral Sequence. Usually it's stated something like this: Let $F\colon \mathcal{A}\to \mathcal{B},G\colon\mathcal{B}\to \mathcal{C}$ be left exact Functors. Assume $F$ sends injective objects to $G$ acyclic objects. And assuming that $\mathcal{A},\mathcal{B}$ have enough injective objects.
Now my question: Is there another version of the Grothendieck Spectral Sequence which doesn't rely on injective objects? I could imagine that the assumption could be restated in a way of assuming that $F$ sends projective objects to something?
Any help is appreciated!