Definition 78.4.1. 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 72.9.1.

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 $\mathcal{P}$.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).