Let $A$ be an abelian category with enough injectives, let $K(A)$ denote the homotopy category (objects = chain complexes, morphisms = chain maps mod chain homotopies), and let $D(A)$ be the derived category (localization of $K(A)$ at the quasi-isomorphisms). Then if $F$ is a left-exact functor $A \to A'$, we speak of the total derived functor $RF: D(A) \to D(A')$ which proceeds by starting with a chain complex, resolving it with an injective complex, and then applying $F$.
Is this total derived functor only really well-defined up to natural isomorphism? After all, it seems to depend on the choice of injective resolution (though different injective resolutions yield quasi-isomorphic images).
It seems to me we could equally well have made this construction on the level of the homotopy category, i.e. make a total derived functor $RF: K(A) \to K(A')$. Why do we pass to the derived category in this construction, where we seemingly lose some information?
