Lemma 21.35.7. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Given K, L, M in D(\mathcal{O}) there is a canonical morphism
K \otimes _\mathcal {O}^\mathbf {L} R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (M, K \otimes _\mathcal {O}^\mathbf {L} L)
in D(\mathcal{O}) 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}-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} \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} \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 21.34.3.
\square
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