Regular tensor product and monoidal structure on Boolean σ-algebras

Determine whether the regular tensor product on Boolean σ-algebras induces a symmetric monoidal structure on the category Boolσ of Boolean algebras with σ-homomorphisms.

Background

While the regular tensor product yields a symmetric monoidal structure on concrete Boolean σ-algebras via Loomis–Sikorski duality, functoriality issues for regular σ-completions obstruct a straightforward extension to all of Boolσ. Clarifying this would unify tensor product treatments across abstract and concrete settings.