The Stacks project

Lemma 73.4.12. Let $S$ be a scheme. Given morphisms $f : X \to Y$, $g : Y \to Z$ in $(\textit{Spaces}/S)_{\acute{e}tale}$ 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 73.4.10. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 73.4.11. $\square$

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