Definition 7.20.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. The functor $u$ is called *cocontinuous* if for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and every covering $\{ V_ j \to u(U)\} _{j \in J}$ of $\mathcal{D}$ there exists a covering $\{ U_ i \to U\} _{i\in I}$ of $\mathcal{C}$ such that the family of maps $\{ u(U_ i) \to u(U)\} _{i \in I}$ refines the covering $\{ V_ j \to u(U)\} _{j \in J}$.

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