Lemma 21.35.8. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Given $K, L$ in $D(\mathcal{O})$ there is a canonical morphism

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

in $D(\mathcal{O})$ functorial in both $K$ and $L$.

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

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

where the first map comes from Lemma 21.34.4. $\square$

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