Lemma 77.16.3. Let $B \to S$ as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let $G/U$ be its stabilizer. Denote $R_ t/U$ the algebraic space $R$ seen as an algebraic space over $U$ via the morphism $t : R \to U$. There is a canonical left action

$a : G \times _ U R_ t \longrightarrow R_ t$

induced by the composition law $c$.

Proof. In terms of points over $T/B$ we define $a(g, r) = c(g, r)$. $\square$

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