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The Stacks project

Lemma 101.39.9. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks which is representable by algebraic spaces. Then the following are equivalent

  1. f satisfies the uniqueness part of the valuative criterion,

  2. for every scheme T and morphism T \to \mathcal{Y} the morphism \mathcal{X} \times _\mathcal {Y} T \to T satisfies the uniqueness part of the valuative criterion as a morphism of algebraic spaces.

Proof. Follows from Lemma 101.39.4 and the definition. \square


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