The Stacks project

Remark 90.8.4. The characterization of smooth morphisms in Remark 90.8.3 is analogous to Schlessinger's notion of a smooth morphism of functors, cf. [Definition 2.2., Sch]. In fact, when $\mathcal{F}$ and $\mathcal{G}$ are cofibered in sets then our notion is equivalent to Schlessinger's. Namely, in this case let $F, G : \mathcal{C}_\Lambda \to \textit{Sets}$ be the corresponding functors, see Remarks 90.5.2 (11). Then $F \to G$ is smooth if and only if for every surjection of rings $B \to A$ in $\mathcal{C}_\Lambda$ the map $F(B) \to F(A) \times _{G(A)} G(B)$ is surjective.