Representable morphisms of presheaves and fibrewise properties

Let $F$ and $G$ be presheaves of sets on $(Sch\mathrm{/}S{)}_{fppf}$ (Fppf sheaves of sets and sheafification) and let $a\text{\hspace{0.17em}⁣}:F\to G$ be a morphism of presheaves (Natural transformation and its components). For a morphism $\xi \text{\hspace{0.17em}⁣}:T\to G$ from an $S$-scheme $T$ — that is, $\xi \in G(T)$, the represented functor of $T$ mapping to $G$ — the fibre product $F{\times }_{G,\xi }T$ is the presheaf $T^{\mathrm{\prime }}⟼\{(x,\phi ):x\in F(T^{\mathrm{\prime }}),\phi \text{\hspace{0.17em}⁣}:T^{\mathrm{\prime }}\to T,a(x)=\xi \circ \phi \}$ with the evident restriction maps (Fibre product of schemes).

The morphism $a$ is representable by schemes when for every $S$-scheme $T$ and every $\xi \text{\hspace{0.17em}⁣}:T\to G$ this fibre product is representable by a scheme (Presheaves, covariantly and contravariantly representable functors, and representations); that is, there is a scheme $U$ and an isomorphism of presheaves $F{\times }_{G,\xi }T\cong {\mathrm{Mor}}_S(-,U)$ (Morphisms of schemes). The representing scheme, when it exists, is well defined up to unique isomorphism, by the Yoneda lemma.

Let $\mathcal{P}$ be a property of morphisms of schemes that is stable under base change. A representable morphism $a$ has property $\mathcal{P}$ when for every $T$ and $\xi$ the induced morphism of schemes $U\to T$ representing the fibre product has property $\mathcal{P}$. Since $\mathcal{P}$ is stable under base change and the construction of the fibre product is compatible with base change in $T$, this is well defined and depends only on $a$. In this way one defines representable etale, flat, surjective, open-immersion and closed-immersion morphisms (Étale morphism of schemes, Flat morphism of schemes, Open immersions of schemes, Closed immersions of schemes).

A morphism of sheaves $a\text{\hspace{0.17em}⁣}:F\to G$ is fppf-locally surjective (or an epimorphism of sheaves) when every section of $G$ lifts fppf-locally to $F$: for every $S$-scheme $T$ and every $\xi \in G(T)$ there is an fppf covering $\{T_i\to T\}$ such that each $\xi {\mathrm{\mid }}_{T_i}$ lies in the image of $a_{T_i}\text{\hspace{0.17em}⁣}:F(T_i)\to G(T_i)$. An etale cover is a representable, etale morphism whose scheme base changes are surjective. For etale morphisms this is equivalent to fppf-local surjectivity: a surjective etale base change is itself an fppf cover and supplies the lift, while local lifts force its image to cover the target. For a general representable morphism, surjectivity on scheme points and fppf-local lifting are distinct notions and must be named separately. These definitions are used only for morphisms of presheaves satisfying the representability clause, so each fibrewise property is a property of actual morphisms of schemes.