Definition 65.5.1. With $S$, and $a : F \to G$ representable as above. Let $\mathcal{P}$ be a property of morphisms of schemes which

is preserved under any base change, see Schemes, Definition 26.18.3, and

is fppf local on the base, see Descent, Definition 35.22.1.

In this case we say that $a$ has *property $\mathcal{P}$* if for every $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and any $\xi \in G(U)$ the resulting morphism of schemes $V_\xi \to U$ has property $\mathcal{P}$.

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