Remark 20.38.10. Let $(X, \mathcal{O}_ X)$ be a ringed space. For $K, K', M, M'$ in $D(\mathcal{O}_ X)$ there is a canonical map

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

Namely, by (20.38.0.1) is the same thing as a map

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

For this we can first flip the middle two factors (with sign rules as in More on Algebra, Section 15.68) and use the maps

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

from Lemma 20.38.5 when thinking of $K = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ X, K)$ and similarly for $K'$, $M$, and $M'$.

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