The Stacks project

Lemma 39.17.3. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer. Denote $R_ t/U$ the scheme $R$ seen as a scheme 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 $T/S$ we define $a(g, r) = c(g, r)$. $\square$

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