Lemma 106.12.2. Let (U, R, s, t, c) be a groupoid in algebraic spaces with s, t : R \to U flat and locally of finite presentation. Consider the algebraic stack \mathcal{X} = [U/R]. Given an algebraic space Y there is a 1-to-1 correspondence between morphisms f : \mathcal{X} \to Y and R-invariant morphisms \phi : U \to Y.
Proof. Criteria for Representability, Theorem 97.17.2 tells us \mathcal{X} is an algebraic stack. Given a morphism f : \mathcal{X} \to Y we let \phi : U \to Y be the composition U \to \mathcal{X} \to Y. Since R = U \times _\mathcal {X} U (Groupoids in Spaces, Lemma 78.22.2) it is immediate that \phi is R-invariant. Conversely, if \phi : U \to Y is an R-invariant morphism towards an algebraic space, we obtain a morphism f : \mathcal{X} \to Y by Groupoids in Spaces, Lemma 78.23.2. You can also construct f from \phi using the explicit description of the quotient stack in Groupoids in Spaces, Lemma 78.24.1. \square
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