Lemma 88.21.8. Let S be a scheme. Let f : X \to Y be a morphism of locally Noetherian formal algebraic spaces which is locally of finite type. Let \{ g_ i : Y_ i \to Y\} be a family of morphisms of formal algebraic spaces which are representable by algebraic spaces and étale such that \coprod g_ i is surjective. Then f is rig-surjective if and only if each f_ i : X \times _ Y Y_ i \to Y_ i is rig-surjective.
Proof. Namely, if f is rig-surjective, so is any base change (Lemma 88.21.4). Conversely, if all f_ i are rig-surjective, so is \coprod f_ i : \coprod X \times _ Y Y_ i \to \coprod Y_ i. By Lemma 88.21.7 the morphism \coprod g_ i : \coprod Y_ i \to Y is rig-surjective. Hence \coprod X \times _ Y Y_ i \to Y is rig-surjective (Lemma 88.21.3). Since this morphism factors through X \to Y we see that X \to Y is rig-surjective by Lemma 88.21.6. \square
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