If I have a filtered object $A = A_0 \supset A_1 \supset A_2 \supset \cdots$ such that the filtration is complete and a left exact functor $F$, is there a spectral sequence with inputs $F(A_i/A_{i+1})$ converging to $F(A)$?
My concern is that since $F$ is merely left exact, the associated graded objects of the induced filtration of $F(A)$ are not simply $F(A_i/A_{i+1})$. Is there some way to account for this defect in a spectral sequence?