The Stacks project

Remark 65.5.2. Consider the property $\mathcal{P}=$“surjective”. In this case there could be some ambiguity if we say “let $F \to G$ be a surjective map”. Namely, we could mean the notion defined in Definition 65.5.1 above, or we could mean a surjective map of presheaves, see Sites, Definition 7.3.1, or, if both $F$ and $G$ are sheaves, we could mean a surjective map of sheaves, see Sites, Definition 7.11.1. If not mentioned otherwise when discussing morphisms of algebraic spaces we will always mean the first. See Lemma 65.5.9 for a case where surjectivity implies surjectivity as a map of sheaves.