Lemma 78.21.1 (046Q)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.21: Functoriality of quotient stacks
Lemma 78.21.1 (cite)

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Lemma 78.21.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ . Let $f : (U, R, s, t, c) \to (U', R', s', t', c')$ be a morphism of groupoids in algebraic spaces over $B$ . Then $f$ induces a canonical $1$ -morphism of quotient stacks

$[f] : [U/R] \longrightarrow [U'/R'].$

Proof. Denote $[U/_{\! p}R]$ and $[U'/_{\! p}R']$ the categories fibred in groupoids over the base site $(\mathit{Sch}/S)_{fppf}$ associated to the functors (78.20.0.1). It is clear that $f$ defines a $1$ -morphism $[U/_{\! p}R] \to [U'/_{\! p}R']$ which we can compose with the stackyfication functor for $[U'/R']$ to get $[U/_{\! p}R] \to [U'/R']$ . Then, by the universal property of the stackyfication functor $[U/_{\! p}R] \to [U/R]$ , see Stacks, Lemma 8.9.2 we get $[U/R] \to [U'/R']$ . $\square$

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