Lemma 21.33.3. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be a morphism of ringed topoi. The relative cup product of Remark 21.19.7 is commutative in the sense that the diagram

$\xymatrix{ Rf_*K \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*L \ar[r] \ar[d]_\psi & Rf_*(K \otimes _\mathcal {O}^\mathbf {L} L) \ar[d]^{Rf_*\psi } \\ Rf_*L \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*K \ar[r] & Rf_*(L \otimes _\mathcal {O}^\mathbf {L} K) }$

is commutative in $D(\mathcal{O}')$ for all $K, L$ in $D(\mathcal{O})$. Here $\psi$ is the commutativity constraint on the derived category (Lemma 21.48.5).

Proof. Omitted. $\square$

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