Lemma 13.31.2. Let $\mathcal{A}$ be an abelian category. Let $I^\bullet $ be a complex. The following are equivalent

$I^\bullet $ is K-injective,

for every quasi-isomorphism $M^\bullet \to N^\bullet $ the map

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(N^\bullet , I^\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(M^\bullet , I^\bullet ) \]is bijective, and

for every complex $N^\bullet $ the map

\[ \mathop{\mathrm{Hom}}\nolimits _{K(\mathcal{A})}(N^\bullet , I^\bullet ) \to \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{A})}(N^\bullet , I^\bullet ) \]is an isomorphism.

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