The Stacks project

Lemma 39.18.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$. Let $G$ be the stabilizer of $(U, R, s, t, c)$ and let $G'$ be the stabilizer of $(U', R', s', t', c')$. Then $G'$ is the base change of $G$ by $g$, i.e., there is a canonical identification $G' = U' \times _{g, U} G$.

Proof. Omitted. $\square$

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