Lemma 13.23.4. Let \mathcal{A} be an abelian category. Assume \mathcal{A} has enough injectives. Then a resolution functor j exists and is unique up to unique isomorphism of functors.
Proof. Consider the set of all objects K^\bullet of K^{+}(\mathcal{A}). (Recall that by our conventions any category has a set of objects unless mentioned otherwise.) By Lemma 13.18.3 every object has an injective resolution. By the axiom of choice we can choose for each K^\bullet an injective resolution i_{K^\bullet } : K^\bullet \to j(K^\bullet ). \square
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