Remark 20.42.11. Let $f : X \to Y$ be a morphism of ringed spaces. Let $K, L$ be objects of $D(\mathcal{O}_ X)$. We claim there is a canonical map
\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (Rf_*L, Rf_*K) \]
Namely, by (20.42.0.1) this is the same thing as a map $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \to Rf_*K$. For this we can use the composition
\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \to Rf_*(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L) \to Rf_*K \]
where the first arrow is the relative cup product (Remark 20.28.7) and the second arrow is $Rf_*$ applied to the canonical map $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_ X}^\mathbf {L} L \to K$ coming from Lemma 20.42.5 (with $\mathcal{O}_ X$ in one of the spots).
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