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).
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