Morphisms of sites can be identified with those geometric morphisms of induced toposes for which the inverse image functor preserves representables.
If both sites have finite limits, then covering-flat functors are precisely the functors that preserve finite limits.
What is the analogue of this story for ∞-sites?
Again, we want to identify morphisms of ∞-sites with those geometric morphisms of induced (nonhypercompleted) ∞-toposes for which the inverse image functor preserves representables.
Thus, a morphism of ∞-sites should be a functor between the underlying ∞-categories that preserves covering families and satisfies the appropriate ∞-analogue of being a covering-flat functor.
What is a covering-flat functor between ∞-sites?
If both ∞-sites admit finite ∞-limits, then covering-flat functors could be defined as finite ∞-limit preserving functors.
However, I am interested in ∞-sites that do not have finite ∞-limits, such as the site of smooth manifolds or the site of cartesian spaces.