The Stacks project

Lemma 39.18.1. 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. Consider the following diagram

\[ \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}& R \times _{s, U} U' \ar[r] \ar[d] & U' \ar[d]^ g \\ U' \times _{U, t} R \ar[d] \ar[r] & R \ar[r]^ s \ar[d]_ t & U \\ U' \ar[r]^ g & U } \]

where all the squares are fibre product squares. Then there is a canonical composition law $c' : R' \times _{s', U', t'} R' \to R'$ such that $(U', R', s', t', c')$ is a groupoid scheme over $S$ and such that $U' \to U$, $R' \to R$ defines a morphism $(U', R', s', t', c') \to (U, R, s, t, c)$ of groupoid schemes over $S$. Moreover, for any scheme $T$ over $S$ the functor of groupoids

\[ (U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c) \]

is the restriction (see above) of $(U(T), R(T), s, t, c)$ via the map $U'(T) \to U(T)$.

Proof. Omitted. $\square$

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