Lemma 87.19.5. Let S be a scheme. Let Y be an affine formal algebraic space over S. Let f : X \to Y be a map of sheaves on (\mathit{Sch}/S)_{fppf} which is representable by algebraic spaces. Then X is a formal algebraic space.
Proof. Write Y = \mathop{\mathrm{colim}}\nolimits Y_\lambda as in Definition 87.9.1. For each \lambda the fibre product X \times _ Y Y_\lambda is an algebraic space. Hence X = \mathop{\mathrm{colim}}\nolimits X \times _ Y Y_\lambda is a formal algebraic space by Lemma 87.13.1. \square
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