The Stacks project

Definition 7.3.1. Let $\mathcal{C}$ be a category, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of presheaves of sets.

1. We say that $\varphi$ is injective if for every object $U$ of $\mathcal{C}$ the map $\varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U)$ is injective.

2. We say that $\varphi$ is surjective if for every object $U$ of $\mathcal{C}$ the map $\varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U)$ is surjective.