The Stacks project

Lemma 88.8.7. Let $\varphi : \mathcal{F} \to \mathcal{G}$ and $\psi : \mathcal{G} \to \mathcal{H}$ be morphisms of categories cofibered in groupoids over $\mathcal{C}_\Lambda $.

  1. If $\varphi $ and $\psi $ are smooth, then $\psi \circ \varphi $ is smooth.

  2. If $\varphi $ is essentially surjective and $\psi \circ \varphi $ is smooth, then $\psi $ is smooth.

  3. If $\mathcal{G}' \to \mathcal{G}$ is a morphism of categories cofibered in groupoids and $\varphi $ is smooth, then $\mathcal{F} \times _\mathcal {G} \mathcal{G}' \to \mathcal{G}'$ is smooth.

Proof. Statements (1) and (2) follow immediately from the definitions. Proof of (3) omitted. Hints: use the formulation of smoothness given in Remark 88.8.3 and use that $\mathcal{F} \times _\mathcal {G} \mathcal{G}'$ is the $2$-fibre product, see Remarks 88.5.2 (13). $\square$


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