Lemma 61.12.17. Given schemes $X$, $Y$, $Y$ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ 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 61.12.15. For the functors on the small sites this follows from the description of the pushforward functors in Lemma 61.12.16. $\square$

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