The Stacks project

Definition 7.42.3. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. We say $u$ is almost cocontinuous if for every object $U$ of $\mathcal{C}$ and every covering $\{ V_ j \to u(U)\} _{j \in J}$ there exists a covering $\{ U_ i \to U\} _{i \in I}$ in $\mathcal{C}$ such that for each $i$ in $I$ we have at least one of the following two conditions

1. $u(U_ i)$ is sheaf theoretically empty, or

2. the morphism $u(U_ i) \to u(U)$ factors through $V_ j$ for some $j \in J$.