Lemma 21.18.4. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi. There is a canonical bifunctorial isomorphism
\[ Lf^*( \mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet ) = Lf^*\mathcal{F}^\bullet \otimes _{\mathcal{O}}^{\mathbf{L}} Lf^*\mathcal{G}^\bullet \]
for $\mathcal{F}^\bullet , \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}'))$.
Proof.
By our construction of derived pullback in Lemma 21.18.2. and the existence of resolutions in Lemma 21.17.11 we may replace $\mathcal{F}^\bullet $ and $\mathcal{G}^\bullet $ by complexes of $\mathcal{O}'$-modules which are K-flat and have flat terms. In this case $\mathcal{F}^\bullet \otimes _{\mathcal{O}'}^{\mathbf{L}} \mathcal{G}^\bullet $ is just the total complex associated to the double complex $\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet $. The complex $\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet )$ is K-flat with flat terms by Lemma 21.17.5 and Modules on Sites, Lemma 18.28.12. Hence the isomorphism of the lemma comes from the isomorphism
\[ \text{Tot}(f^*\mathcal{F}^\bullet \otimes _{\mathcal{O}} f^*\mathcal{G}^\bullet ) \longrightarrow f^*\text{Tot}(\mathcal{F}^\bullet \otimes _{\mathcal{O}'} \mathcal{G}^\bullet ) \]
whose constituents are the isomorphisms $f^*\mathcal{F}^ p \otimes _{\mathcal{O}} f^*\mathcal{G}^ q \to f^*(\mathcal{F}^ p \otimes _{\mathcal{O}'} \mathcal{G}^ q)$ of Modules on Sites, Lemma 18.26.2.
$\square$
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