The Stacks project

Remark 13.23.7. Suppose that $\mathcal{A}$ is a “big” abelian category with enough injectives such as the category of abelian groups. In this case we have to be slightly more careful in constructing our resolution functor since we cannot use the axiom of choice with a quantifier ranging over a class. But note that the proof of the lemma does show that any two localization functors are canonically isomorphic. Namely, given quasi-isomorphisms $i : K^\bullet \to I^\bullet $ and $i' : K^\bullet \to J^\bullet $ of a bounded below complex $K^\bullet $ into bounded below complexes of injectives there exists a unique(!) morphism $a : I^\bullet \to J^\bullet $ in $K^{+}(\mathcal{I})$ such that $i' = i \circ a$ as morphisms in $K^{+}(\mathcal{I})$. Hence the only issue is existence, and we will see how to deal with this in the next section.