Lemma 21.18.8. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{K}^\bullet$ and $\mathcal{M}^\bullet$ be complexes of $\mathcal{O}_\mathcal {D}$-modules. The diagram

$\xymatrix{ Lf^*(\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {D}}^\mathbf {L} \mathcal{M}^\bullet ) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{M}^\bullet ) \ar[d] \\ Lf^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} Lf^*\mathcal{M}^\bullet \ar[d] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{M}^\bullet ) \ar[d] \\ f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {C}}^\mathbf {L} f^*\mathcal{M}^\bullet \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{M}^\bullet ) }$

commutes.

Proof. We will use the existence of K-flat resolutions with flat terms (Lemma 21.17.11), we will use that derived pullback is computed by such complexes (Lemma 21.18.2), and that pullbacks preserve these properties (Lemma 21.18.1). If we choose such resolutions $\mathcal{P}^\bullet \to \mathcal{K}^\bullet$ and $\mathcal{Q}^\bullet \to \mathcal{M}^\bullet$, then we see that

$\xymatrix{ Lf^*\text{Tot}(\mathcal{P}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{Q}^\bullet ) \ar[r] \ar[d] & Lf^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{M}^\bullet ) \ar[d] \\ f^*\text{Tot}(\mathcal{P}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{Q}^\bullet ) \ar[d] \ar[r] & f^*\text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {D}} \mathcal{M}^\bullet ) \ar[d] \\ \text{Tot}(f^*\mathcal{P}^\bullet \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{Q}^\bullet ) \ar[r] & \text{Tot}(f^*\mathcal{K}^\bullet \otimes _{\mathcal{O}_\mathcal {C}} f^*\mathcal{M}^\bullet ) }$

commutes. However, now the left hand side of the diagram is the left hand side of the diagram by our choice of $\mathcal{P}^\bullet$ and $\mathcal{Q}^\bullet$ and Lemma 21.17.5. $\square$

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