Lemma 101.38.7. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks which is representable by algebraic spaces and separated. Let \mathcal{V} \subset \mathcal{Y} be an open substack such that \mathcal{V} \to \mathcal{Y} is quasi-compact. Let s : \mathcal{V} \to \mathcal{X} be a morphism such that f \circ s = \text{id}_\mathcal {V}. Let \mathcal{Y}' be the scheme theoretic image of s. Then \mathcal{Y}' \to \mathcal{Y} is an isomorphism over \mathcal{V}.
Proof. By Lemma 101.7.7 the morphism s : \mathcal{V} \to \mathcal{Y} is quasi-compact. Hence the construction of the scheme theoretic image \mathcal{Y}' of s commutes with flat base change by Lemma 101.38.5. Thus to prove the lemma we may assume \mathcal{Y} is representable by an algebraic space and we reduce to the case of algebraic spaces which is Morphisms of Spaces, Lemma 67.16.7. \square
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