Let us consider the category of modal spaces, whose objects are pair $(X,R)$ with $X$ a Stone space and $R$ a closed binary relation on $X$ such that $$ \lbrace x \mid \exists y \in O : x \mathrel{R} y \rbrace $$ is clopen for every clopen $O$ and whose morphism are continuous functions $f$ such that
- $x \mathrel{R} y$ implies $f(x) \mathrel{R} f(y)$,
- $f(x) \mathrel{R} y$ implies there exists $z$ such that $x \mathrel{R} z$ and $f(z) = y$.
I wanted to know if there exist a nice description of the coproduct of an infinite family $(X_i,R_i)$ in this category and where to find this description, I'm somehow thinking of $(\beta(\coprod X_i), (\coprod R_i)^-)$, with $\beta(\coprod X_i)$ the Stone-Cech compactification of the disjoint union and with $(\coprod R_i)^-$ the closure of $\coprod R_i$ in $(\beta(\coprod X_i))^2$.