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)