Before revision:  In doing so, we shift the focus from dagger-special commutative Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma comagma pair. Thus this work is of foundational interest as update structures form a strictly more general class of algebraic objects, the taming of which promises to illuminate novel relationships between separately studied mathematical structures .
After revision:  In doing so, we shift the focus from dagger-special commutative Frobenius algebras to interacting (co)magma (co)module pairs, showing that the algebraic properties of the (co)multiplication arise from the module-comodule interaction, rather than direct assumptions about the magma-comagma pair. We then begin to investigate the zoo of possible update structures, introducing the notions of classical security-flagged databases, and databases of quantum systems. This work is of foundational interest as update structures form a strictly more general class of algebraic objects, the taming of which promises to illuminate novel relationships between separately studied mathematical structures .