Lemma 96.3.2. Let $S$ be a locally Noetherian scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Assume either

$F$ is formally smooth on objects (Criteria for Representability, Section 95.6),

$F$ is representable by algebraic spaces and formally smooth, or

$F$ is representable by algebraic spaces and smooth.

Then for every finite type field $k$ over $S$ and object $x_0$ of $\mathcal{X}$ over $k$ the functor (96.3.1.1) is smooth in the sense of Formal Deformation Theory, Definition 88.8.1.

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