Lemma 89.26.1. Let $\mathcal{F}$ be a category cofibered in groupoids over $\mathcal{C}_\Lambda $. Let $U : \mathcal{C}_\Lambda \to \textit{Sets}$ be a functor. Let $f : U \to \mathcal{F}$ be a smooth morphism of categories cofibered in groupoids. Then:

If $(U, R, s, t, c)$ is the groupoid in functors on $\mathcal{C}_\Lambda $ constructed from $f : U \to \mathcal{F}$ in Lemma 89.25.2, then $(U, R, s, t, c)$ is smooth.

If $f : U(k) \to \mathcal{F}(k)$ is essentially surjective, then the morphism $[f] : [U/R] \to \mathcal{F}$ of Lemma 89.25.3 is an equivalence.

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