Let X be a topological space and $A$ be a ring. Suppose we have an exact sequence of sheaves of $A$-modules on X, $0\longrightarrow F\longrightarrow G\longrightarrow H\longrightarrow 0$, suppose we are given a left-exact functor $\phi$ in the category of sheaves of $A$-modules, and $R\phi$ its derived functor. Suppose now $R\phi(G),R\phi(H)$ are computable. Is there a general method to compute cohomology $R\phi(F)$ from that of $R\phi(G),R\phi(H)$ ?
In the derived category $F$ is qis to the complex $G\longrightarrow H$, then I tried to compute by hand the resolution of this complex and try to get some information applying the $R\phi$ functor. In the case I am working on, it was sufficient, but I am wondering if there is a general method.