Definition 88.8.1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a morphism of categories cofibered in groupoids over $\mathcal{C}_\Lambda$. We say $\varphi$ is smooth if it satisfies the following condition: Let $B \to A$ be a surjective ring map in $\mathcal{C}_\Lambda$. Let $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{G}(B)), x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(A))$, and $y \to \varphi (x)$ be a morphism lying over $B \to A$. Then there exists $x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{F}(B))$, a morphism $x' \to x$ lying over $B \to A$, and a morphism $\varphi (x') \to y$ lying over $\text{id}: B \to B$, such that the diagram

$\xymatrix{ \varphi (x') \ar[r] \ar[dr] & y \ar[d] \\ & \varphi (x) }$

commutes.

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