Lemma 78.29.1. Let $B \to S$ be as in Section 78.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be the stabilizer group algebraic space. The commutative diagram
\[ \xymatrix{ R \ar[d]^{\Delta _{R/U \times _ B U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times _{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times _{(U \times _ B U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times _{s, U} G \ar[r] & G } \]
the two left horizontal arrows are isomorphisms and the right square is a fibre product square.
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