Suppose we have two abelian categories $A$ and $B$. Assume that $A$ has enough injectives. Now consider a sequence of functors $F_0,F_1,F_2,....$.
Such that a short exact sequence in $A$ induces a long exact sequence in terms of $F_i$. Can we then claim that $F_i$ are the right derived functors of the functor $F_0$?