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

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

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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