Lemma 78.17.1. Let B \to S as in Section 78.3. Let (U, R, s, t, c) be a groupoid in algebraic spaces over B. Let g : U' \to U be a morphism of algebraic spaces. 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 in algebraic spaces over B and such that U' \to U, R' \to R defines a morphism (U', R', s', t', c') \to (U, R, s, t, c) of groupoids in algebraic spaces over B. Moreover, for any scheme T over B the functor of groupoids
(U'(T), R'(T), s', t', c') \to (U(T), R(T), s, t, c)
is the restriction (see Groupoids, Section 39.18) of (U(T), R(T), s, t, c) via the map U'(T) \to U(T).
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