Remark 21.34.10. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $K, L$ be objects of $D(\mathcal{O}_\mathcal {C})$. 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 (21.34.0.1) this is the same thing as a map $Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L \to Rf_*K$. For this we can use the composition

$Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} Rf_*L \to Rf_*(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L) \to Rf_*K$

where the first arrow is the relative cup product (Remark 21.20.6) and the second arrow is $Rf_*$ applied to the canonical map $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L, K) \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} L \to K$ coming from Lemma 21.34.6 (with $\mathcal{O}_\mathcal {C}$ in one of the spots).

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