Lemma 77.11.6. Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $B' \to B$ be a morphism of algebraic spaces. Then the base changes $U' = B' \times _ B U$, $R' = B' \times _ B R$ endowed with the base changes $s'$, $t'$, $c'$ of the morphisms $s, t, c$ form a groupoid in algebraic spaces $(U', R', s', t', c')$ over $B'$ and the projections determine a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoids in algebraic spaces over $B$.

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

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