Lemma 20.42.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given $K, L, M$ in $D(\mathcal{O}_ X)$ there is a canonical morphism

$K \otimes _{\mathcal{O}_ X}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$

in $D(\mathcal{O}_ X)$ functorial in $K, L, M$.

Proof. Choose a K-flat complex $\mathcal{K}^\bullet$ representing $K$, and a K-injective complex $\mathcal{I}^\bullet$ representing $L$, and choose any complex of $\mathcal{O}_ X$-modules $\mathcal{M}^\bullet$ representing $M$. Choose a quasi-isomorphism $\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet ) \to \mathcal{J}^\bullet$ where $\mathcal{J}^\bullet$ is K-injective. Then we use the map

$\text{Tot}\left( \mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{I}^\bullet ) \right) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet )) \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet (\mathcal{M}^\bullet , \mathcal{J}^\bullet )$

where the first map is the map from Lemma 20.41.3. $\square$

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