Remark 20.28.7. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. The adjointness of $Lf^*$ and $Rf_*$ allows us to construct a relative cup product

$Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L \longrightarrow Rf_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} L)$

in $D(\mathcal{O}_ Y)$ for all $K, L$ in $D(\mathcal{O}_ X)$. Namely, this map is adjoint to a map $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ for which we can take the composition of the isomorphism $Lf^*(Rf_*K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L) = Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L$ (Lemma 20.27.3) with the map $Lf^*Rf_*K \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*Rf_*L \to K \otimes _{\mathcal{O}_ X}^\mathbf {L} L$ coming from the counit $Lf^* \circ Rf_* \to \text{id}$.

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