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 that 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)