The category of propositional deductive systems

We define the category $\text{QM}$ of quantales and their modules and prove the existence of coproducts, and the existence of pushout and amalgamated coproducts under certain conditions. Then we define the non-full subcategory ${\text{DS}}_0$ of propositional deductive systems, show that it is equivalent to the one of real'' propositional logics whose morphisms are interpretations (modulo a language translation, when needed), and prove that the coproduct in $\DS_0$ is precisely the deductive system called logical coproduct’’ in \cite{ruslu}. Last, we discuss amalgamation in ${\text{DS}}_0$.