The Stacks project

Lemma 79.12.9. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } \]

be a fibre product square of algebraic spaces over $S$. Then

\[ \xymatrix{ (X'/Y')_{fin} \ar[d] \ar[r] & (X/Y)_{fin} \ar[d] \\ Y' \ar[r] & Y } \]

is a fibre product square of sheaves on $(\mathit{Sch}/S)_{fppf}$.

Proof. It follows immediately from the definitions that the sheaf $(X'/Y')_{fin}$ is equal to the sheaf $Y' \times _ Y (X/Y)_{fin}$. $\square$

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