The lattice of smooth sublocales as a Bruns-Lakser completion

We characterise the frame morphisms $f:L\to M$ that lift to frame maps $\bar{f}:{\mathsf{S}}_b(L)\to {\mathsf{S}}_b(M)$, where ${\mathsf{S}}_b(L)$ is the collection of joins of complemented sublocales of a frame $L$, or equivalently the Booleanization of the collection $\mathsf{S}(L)$ of all its sublocales. We do so by proving that ${\mathsf{S}}_b(L)$ is isomorphic to the Bruns–Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form $\mathfrak{c}(a)\cap \mathfrak{o}(b)$ for $a,b\in L$.