Jònsson and Tarski showed a generalization of the Stone representation theorem: they showed how to construct, given $A$ a Boolean algebra with operators (BAO), a complete atomic completely additive BAO $A^\sigma$ such that $A$ is a subalgebra of $A^\sigma$.
My question is: is $A^\sigma$ special among those complete atomic completely additive BAOs $B$ that $A$ embeds into? I know that $(\cdot)^\sigma$ is functorial, but is it special among other endofunctors over the category of BAOs (e.g., does it satisfy some universal property)? If I already know that $A$ embeds into a complete atomic completely additive $B$, what can I know about the relationship between $A^\sigma$ and $B$?