Lemma 20.37.7. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(\mathcal{I}')^\bullet \to \mathcal{I}^\bullet$ be a quasi-isomorphism of K-injective complexes of $\mathcal{O}_ X$-modules. Let $(\mathcal{L}')^\bullet \to \mathcal{L}^\bullet$ be a quasi-isomorphism of complexes of $\mathcal{O}_ X$-modules. Then

$\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet )$

is a quasi-isomorphism.

Proof. Let $M$ be the object of $D(\mathcal{O}_ X)$ represented by $\mathcal{I}^\bullet$ and $(\mathcal{I}')^\bullet$. Let $L$ be the object of $D(\mathcal{O}_ X)$ represented by $\mathcal{L}^\bullet$ and $(\mathcal{L}')^\bullet$. By Lemma 20.37.6 we see that the sheaves

$H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{L}^\bullet , (\mathcal{I}')^\bullet )) \quad \text{and}\quad H^0(\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ((\mathcal{L}')^\bullet , \mathcal{I}^\bullet ))$

are both equal to the sheaf associated to the presheaf

$U \longmapsto \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(L|_ U, M|_ U)$

Thus the map is a quasi-isomorphism. $\square$

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