The Stacks project

Lemma 106.8.5. 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. $f$ is formally smooth,

  2. for every scheme $T$ and morphism $T \to \mathcal{Y}$ the morphism $\mathcal{X} \times _\mathcal {Y} T \to T$ is formally smooth as a morphism of algebraic spaces.

Proof. Follows from Categories, Lemma 4.44.2 and the definition. $\square$

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