Question: What are some "surprising" examples of functors (resp. $\infty$-functors) $F$ which preserve cofiltered limits?
I'm not quite sure what "surprising" means, but I think that
"Surprising" should probably mean that $F$ doesn't preserve all connected limits (resp. all contractible limits) -- though one of my main examples below seems to violate this, so take it with a grain of salt.
"Surprising" should also mean that $F$ is not just obtained as $G^{op}$ where $G$ is a functor which preserves filtered colimits for some familiar reason.
I've learned of two interesting examples from Joyal's paper on combinatorial species (Theorem 1 in the appendix). Namely,
In $Set$ or $Spaces$, arbitrary coproducts commute with cofiltered limits. I believe this generalizes to any topos or $\infty$-topos, but I'm not quite sure.
In $Set$ or $Spaces$, quotients by a finite group action commute with cofiltered limits. Actually, I think this generalizes to an arbitrary action of a group object in $Spaces$, and to all contractible limits, so maybe it's not actually "surprising" according to (1) above. Nonetheless, I'm still surprised by it since it's a colimit construction, so I'll count it. I'm not quite sure if this generalizes to an arbitrary $\infty$-topos.
