Lemma 13.23.3. Let $\mathcal{A}$ be an abelian category with enough injectives. Given a resolution functor $(j, i)$ there is a unique way to turn $j$ into a functor and $i$ into a $2$-isomorphism producing a $2$-commutative diagram

\[ \xymatrix{ K^{+}(\mathcal{A}) \ar[rd] \ar[rr]_ j & & K^{+}(\mathcal{I}) \ar[ld] \\ & D^{+}(\mathcal{A}) } \]

where $\mathcal{I}$ is the full additive subcategory of $\mathcal{A}$ consisting of injective objects.

**Proof.**
For every morphism $\alpha : K^\bullet \to L^\bullet $ of $K^{+}(\mathcal{A})$ there is a unique morphism $j(\alpha ) : j(K^\bullet ) \to j(L^\bullet )$ in $K^{+}(\mathcal{I})$ such that

\[ \xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_{i_{K^\bullet }} & L^\bullet \ar[d]^{i_{L^\bullet }} \\ j(K^\bullet ) \ar[r]^{j(\alpha )} & j(L^\bullet ) } \]

is commutative in $K^{+}(\mathcal{A})$. To see this either use Lemmas 13.18.6 and 13.18.7 or the equivalent Lemma 13.18.8. The uniqueness implies that $j$ is a functor, and the commutativity of the diagram implies that $i$ gives a $2$-morphism which witnesses the $2$-commutativity of the diagram of categories in the statement of the lemma.
$\square$

## Comments (1)

Comment #8420 by ElĂas Guisado on