Lemma 39.17.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $G$ be the stabilizer group scheme of $R$. Let

as a group scheme over $U \times _ S U$. The action of $G$ on $R$ of Lemma 39.17.3 induces an action of $G_0$ on $R$ over $U \times _ S U$ which turns $R$ into a pseudo $G_0$-torsor over $U \times _ S U$.

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