A double functor is a functor between double categories. Since if and are double categories, the only possible sort of functor is a double functor, it is unambiguous to leave off the adjective and simply speak about “functors” between double categories.
However, just like 2-functors, double functors do come in different flavors: strict, pseudo/strong, lax, and oplax. Moreover, these various flavors can be chosen more or less independently in the two directions of a double category (vertical and horizontal). Thus we can have functors which are strict in both directions, strict in one direction and pseudo in the other, pseudo in both directions, strict in one direction and lax in the other, and so on. (It’s not clear whether lax+lax or lax+oplax are sensible, though.)
If and are strict double categories, i.e. internal categories in , then a strict double functor is simply an internal functor in . Thus, it takes objects, arrows of both sorts, and squares in to the same structures in , preserving sources and targets and also preserving all identities and composites.