Lemma 21.33.2. 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 associative in the sense that the diagram

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

is commutative in $D(\mathcal{O}')$ for all $K, L, M$ in $D(\mathcal{O})$.

**Proof.**
Going around either side we obtain the map adjoint to the obvious map

\begin{align*} Lf^*(Rf_*K \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*L \otimes _{\mathcal{O}'}^\mathbf {L} Rf_*M) & = Lf^*(Rf_*K) \otimes _\mathcal {O}^\mathbf {L} Lf^*(Rf_*L) \otimes _\mathcal {O}^\mathbf {L} Lf^*(Rf_*M) \\ & \to K \otimes _\mathcal {O}^\mathbf {L} L \otimes _\mathcal {O}^\mathbf {L} M \end{align*}

in $D(\mathcal{O})$.
$\square$

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