nLab cover-preserving functor

If $C$ and $D$ are sites, a functor $F\colon C\to D$ is cover-preserving (some people say continuous) if whenever $R$ is a covering family of an object $U\in C$, its image $F(R)$ is a covering family of $F(U)$ in $D$.

If $F$ is also flat, then it is called a morphism of sites, and induces a geometric morphism between sheaf toposes.