The Stacks project

Remark 89.8.3. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda $. Let $B \to A$ be a ring map in $\mathcal{C}_\Lambda $. Choices of pushforwards along $B \to A$ for objects in the fiber categories $\mathcal{F}(B)$ and $\mathcal{G}(B)$ determine functors $\mathcal{F}(B) \to \mathcal{F}(A)$ and $\mathcal{G}(B) \to \mathcal{G}(A)$ fitting into a $2$-commutative diagram

\[ \xymatrix{ \mathcal{F}(B) \ar[r]^{\varphi } \ar[d] & \mathcal{G}(B) \ar[d] \\ \mathcal{F}(A) \ar[r]^{\varphi } & \mathcal{G}(A) . } \]

Hence there is an induced functor $\mathcal{F}(B) \to \mathcal{F}(A) \times _{\mathcal{G}(A)} \mathcal{G}(B)$. Unwinding the definitions shows that $\varphi : \mathcal{F} \to \mathcal{G}$ is smooth if and only if this induced functor is essentially surjective whenever $B \to A$ is surjective (or equivalently, by Lemma 89.8.2, whenever $B \to A$ is a small extension).

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