When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived categories). It holds when $F$ sends some adopted to $F$ class into an adopted to $G$ class.
My question is whether this is only technical issue, or there is something meaningful behind. For example, is there a situation when this does not hold, and somehow thinking that it does hold give wrong answers. Or, is there a more general context in which it always holds.