Lemma 39.13.6. Let $(U, R, s, t, c)$ be a groupoid over a scheme $S$. Let $S' \to S$ be a morphism. Then the base changes $U' = S' \times _ S U$, $R' = S' \times _ S R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid scheme $(U', R', s', t', c')$ over $S'$ and the projections determine a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$.

**Proof.**
Omitted. Hint: $R' \times _{s', U', t'} R' = S' \times _ S (R \times _{s, U, t} R)$.
$\square$

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