Lemma 39.22.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be the stabilizer group scheme. The commutative diagram

the two left horizontal arrows are isomorphisms and the right square is a fibre product square.

Lemma 39.22.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. Let $G \to U$ be the stabilizer group scheme. The commutative diagram

\[ \xymatrix{ R \ar[d]^{\Delta _{R/U \times _ S U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times _{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times _{(U \times _ S U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times _{s, U} G \ar[r] & G } \]

the two left horizontal arrows are isomorphisms and the right square is a fibre product square.

**Proof.**
Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry.
$\square$

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