Definition 39.21.1. Let $S$ be a scheme. Let $f : (U', R', s', t', c') \to (U, R, s, t, c)$ be a morphism of groupoid schemes over $S$. We say $f$ is cartesian, or that $(U', R', s', t', c')$ is cartesian over $(U, R, s, t, c)$, if the diagram

$\xymatrix{ R' \ar[r]_ f \ar[d]_{s'} & R \ar[d]^ s \\ U' \ar[r]^ f & U }$

is a fibre square in the category of schemes. A morphism of groupoid schemes cartesian over $(U, R, s, t, c)$ is a morphism of groupoid schemes compatible with the structure morphisms towards $(U, R, s, t, c)$.

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