TITLE: Finite coproducts commute with which limits in Set?
QUESTION [7 upvotes]: It is well known that (small) coproducts commute with connected limits in $\mathbf{Set}$. With which class of limits do finite coproducts commute?
Ideally, we should furthermore like to know whether the class of finite coproducts is closed [1] in the sense that the class of finite coproducts is precisely the class commuting with the given class of limits in $\mathbf{Set}$.
[1] Notes on Commutation of Limits and Colimits, Bjerrum–Johnstone–Leinster–Sawin (2015)

REPLY [11 votes]: Indeed, let $D$ be a category. The canonical functor $D \to \pi_0(D)$ is both cofinal and coinitial. Therefore, if finite coproducts commute with $D$-limits in a category $\mathcal C$, then finite coproducts commute with $\pi_0(D)$-limits. And it is easily seen that the only discrete limit shapes with which finite coproducts commute in $Set$ are the singleton ones. So as Tom Goodwillie supposed, the only limit shapes with with finite coproducts commute in $Set$ are the connected ones.
Finite coproducts are not closed -- they don't include splitting of idempotents, which commutes with any limit whatsoever. But I believe that the finite disjoint unions of absolute colimit shapes do form a closed class.