Finite coproducts, coregularity and coexactness for profinite interior algebras

In previous articles, we showed that the category of profinite $L$-algebras (where $L$ is a normal modal logic with the finite model property) is monadic over $\mathbf{\text{Set}}$. Then, we developed sequent calculi for extensions of the language of $L$ with infinitary conjunctions and disjunctions, proving completeness with respect to profinite $L$-algebras and relating syntactic properties of the calculi with regularity/exactness properties of the category opposite to profinite $L$-algebras. In this paper, we focus on the algebraic perspective: we characterize those $L$ extending $S4$ whose profinite algebras enjoy such categorical properties.