Lemma 21.18.3. Consider morphisms of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ and $g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{E}), \mathcal{O}_\mathcal {E})$. Then $Lf^* \circ Lg^* = L(g \circ f)^*$ as functors $D(\mathcal{O}_\mathcal {E}) \to D(\mathcal{O}_\mathcal {C})$.

Proof. Let $E$ be an object of $D(\mathcal{O}_\mathcal {E})$. By construction $Lg^*E$ is computed by choosing a complex $\mathcal{K}^\bullet$ as in Lemma 21.18.1 representing $E$ and setting $Lg^*E = g^*\mathcal{K}^\bullet$. By transitivity of pullback functors the complex $g^*\mathcal{K}^\bullet$ pulled back by any morphism of ringed topoi $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ is K-flat. Hence $g^*\mathcal{K}^\bullet$ is a complex as in Lemma 21.18.1 representing $Lg^*E$. We conclude $Lf^*Lg^*E$ is given by $f^*g^*\mathcal{K}^\bullet = (g \circ f)^*\mathcal{K}^\bullet$ which also represents $L(g \circ f)^*E$. $\square$

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