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The Stacks project

Lemma 97.11.4. Let S be a scheme. Let X \to Z \to B be morphisms of algebraic spaces over S. The following diagram

\xymatrix{ \mathit{Mor}_ B(Z, X) \ar[r] & \mathit{Mor}_ B(Z, Z) \\ \text{Res}_{Z/B}(X) \ar[r] \ar[u] & B \ar[u]_{\text{id}_ Z} }

is a cartesian diagram of sheaves on (\mathit{Sch}/S)_{fppf}.

Proof. Omitted. Hint: Exercise in the functorial point of view in algebraic geometry. \square

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