Lemma 18.13.3 (03D8)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.13: Morphisms of ringed topoi and modules
Lemma 18.13.3 (cite)

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Lemma 18.13.3. $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. There are canonical isomorphisms of functors $(g \circ f)_* \cong g_* \circ f_*$ and $(g \circ f)^* \cong f^* \circ g^*$ .

Proof. This is clear from the definitions. $\square$

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