Lemma 34.4.18 (021J)—The Stacks project

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Part 2: Schemes
Chapter 34: Topologies on Schemes
Section 34.4: The étale topology
Lemma 34.4.18 (cite)

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Lemma 34.4.18. Given schemes $X$ , $Y$ , $Y$ in $\mathit{Sch}_{\acute{e}tale}$ and morphisms $f : X \to Y$ , $g : Y \to Z$ we have $g_{big} \circ f_{big} = (g \circ f)_{big}$ and $g_{small} \circ f_{small} = (g \circ f)_{small}$ .

Proof. This follows from the simple description of pushforward and pullback for the functors on the big sites from Lemma 34.4.16. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 34.4.17. $\square$

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