The Stacks project

Lemma 18.40.10. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ 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 locally ringed topoi. Then the composition $(g, g^\sharp ) \circ (f, f^\sharp )$ (see Definition 18.7.1) is also a morphism of locally ringed topoi.

Proof. Omitted. $\square$

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