Lemma 13.19.8. Let $\mathcal{A}$ be an abelian category. Let $P^\bullet $ be bounded above complex consisting of projective objects. Let $L^\bullet \in K(\mathcal{A})$. Then

\[ \mathop{\mathrm{Mor}}\nolimits _{K(\mathcal{A})}(P^\bullet , L^\bullet ) = \mathop{\mathrm{Mor}}\nolimits _{D(\mathcal{A})}(P^\bullet , L^\bullet ). \]

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