Let $$0 \to A^\bullet \to B^\bullet \to C^\bullet \to 0$$ be a short exact sequence of bounded complexes of objects in an abelian category with enough injectives. Does there exist Cartan-Eilenberg resolutions $A^\bullet \hookrightarrow I^{\bullet,\bullet}$, $B^\bullet \hookrightarrow J^{\bullet,\bullet}$, and $C^\bullet \hookrightarrow K^{\bullet,\bullet}$ such that $$0 \to I^{\bullet,\bullet}\to J^{\bullet,\bullet} \to K^{\bullet,\bullet} \to 0$$ is exact extending the above short exact sequence?
This is probably well-known and I would like to know in which references we can cite this result.