Recall that a morphism of sites is a covering-flat functor that preserves covering families.

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.

allpullbacks are assumed, only pullbacks along covers. so the site of smooth manifolds is okay here, right? anyway, I agree the theory in that appendix is probably not quite what you want, and only mention it because I thought it may still be helpful. $\endgroup$ – Dylan Wilson Mar 21 at 11:56