  We build upon our recently introduced concept of an update structure to show
that it is a generalisation of very-well-behaved lenses, that is, there is a
bijection between a strict subset of update structures and vwb lenses in
cartesian categories. We show that update structures are also sufficiently
general to capture quantum observables, pinpointing the additional assumptions
required to make the two coincide. In doing so, we shift the focus from special
commutative dagger-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 place previously distinct areas of research in a general
class of operationally motivated structures, we expect the taming of this class
to illuminate novel relationships between separately studied topics in computer
science, physics and mathematics.
