Definition 7.13.1. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. A functor $u : \mathcal{C} \to \mathcal{D}$ is called *continuous* if for every $\{ V_ i \to V\} _{i\in I} \in \text{Cov}(\mathcal{C})$ we have the following

$\{ u(V_ i) \to u(V)\} _{i\in I}$ is in $\text{Cov}(\mathcal{D})$, and

for any morphism $T \to V$ in $\mathcal{C}$ the morphism $u(T \times _ V V_ i) \to u(T) \times _{u(V)} u(V_ i)$ is an isomorphism.

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