Lemma 39.17.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The scheme $G$ defined by the cartesian square

is a group scheme over $U$ with composition law $m$ induced by the composition law $c$.

Lemma 39.17.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The scheme $G$ defined by the cartesian square

\[ \xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta } & U \times _ S U } \]

is a group scheme over $U$ with composition law $m$ induced by the composition law $c$.

**Proof.**
This is true because in a groupoid category the set of self maps of any object forms a group.
$\square$

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