The Stacks project

Definition 80.4.1. Let $S$ be a scheme. Let $a : F \to G$ be a map of presheaves on $(\mathit{Sch}/S)_{fppf}$ which is representable by algebraic spaces. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which

1. is preserved under any base change, and

2. is fppf local on the base, see Descent on Spaces, Definition 74.10.1.

In this case we say that $a$ has property $\mathcal{P}$ if for every scheme $U$ and $\xi : U \to G$ the resulting morphism of algebraic spaces $U \times _ G F \to U$ has property $\mathcal{P}$.