The Stacks project

Lemma 87.25.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent

1. $f$ is surjective,

2. for every scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a surjective morphism of formal algebraic spaces,

3. for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a surjective morphism of formal algebraic spaces,

4. there exists a covering $\{ Y_ j \to Y\}$ as in Definition 87.11.1 such that each $X \times _ Y Y_ j \to Y_ j$ is a surjective morphism of formal algebraic spaces,

5. there exists a surjective morphism $Z \to Y$ of formal algebraic spaces such that $X \times _ Y Z \to Z$ is surjective, and

6. add more here.