It is well known that (small) coproducts [commute with connected limits](https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#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](https://arxiv.org/abs/1409.7860), Bjerrum–Johnstone–Leinster–Sawin (2015)



Finite coproducts commute with which limits in Set?
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.


