The Stacks project

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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