Lemma 89.8.7. 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 $.

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

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

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.

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