The Stacks project

Definition 96.3.1. Let $S$ be a scheme. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $f$ is limit preserving if for every directed limit $U = \mathop{\mathrm{lim}}\nolimits U_ i$ of affine schemes over $S$ the diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{U_ i} \ar[r] \ar[d]_ f & \mathcal{X}_ U \ar[d]^ f \\ \mathop{\mathrm{colim}}\nolimits \mathcal{Y}_{U_ i} \ar[r] & \mathcal{Y}_ U } \]

of fibre categories is $2$-cartesian.

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