Lemma 21.33.5. Consider a commutative square
\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}_{\mathcal{C}'}) \ar[r]_{g'} \ar[d]_{f'} & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}, \mathcal{O}_{\mathcal{C}}) \ar[d]^ f \\ (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}'), \mathcal{O}_{\mathcal{D}'}) \ar[r]^ g & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_{\mathcal{D}}) } \]
of ringed topoi. Let $K, L$ in $D(\mathcal{O}_{\mathcal{C}})$. The relative cup product is compatible with the square in the sense that the diagram
\[ \xymatrix{ Lg^*(Rf_*K \otimes _{\mathcal{O}_{\mathcal{D}}}^\mathbf {L} Rf_*L) \ar[r] \ar@{=}[d] & Lg^*(Rf_*(K \otimes _{\mathcal{O}_{\mathcal{C}}}^\mathbf {L} L)) \ar[d] \\ Lg^*Rf_*K \otimes _{\mathcal{O}_{\mathcal{D}'}}^\mathbf {L} Lg^*Rf_*L \ar[d] & R(f')_*L(g')^*(K \otimes _{\mathcal{O}_{\mathcal{C}}}^\mathbf {L} L) \ar@{=}[d] \\ R(f')_*(L(g')^*K \otimes _{\mathcal{O}_{\mathcal{D}'}} R(f')_*(L(g')^*L \ar[r] & R(f')_*(L(g')^*K \otimes _{\mathcal{O}_{\mathcal{C}'}}^\mathbf {L} L(g')^*L) } \]
is commutative in $D(\mathcal{O}_{\mathcal{D}'})$. The horizontal arrows are given by the relative cup product (Remark 21.19.7) and the vertical arrows are given by the base change map (Remark 21.19.3) and Lemma 21.18.4.
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