A new advancement is presented in a broad ongoing effort to develop Dynamical systems theory in the language of Category theory. A new idea will be presented to describe a general dynamical systems as an enriched functor, and change of variables as enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics – a semigroup action on the domain; a parameterized family of endomorphisms; and a transformation of time-space into the collection of endomorphisms. A collection of categorical axioms are presented that provides a complete categorical language to develop dynamical systems theory. True to the philosophy of dynamical systems, none of these assumptions are rooted in specific contexts such as topology and measure spaces. The equivalence of the three descriptions is further used to construct other related notions,such as transfer operators, orbits and sub-shifts. All of these objects are defined by their structural role and universal properties, instead of their usual pointwise definitions.

Source : https://arxiv.org/pdf/2509.05900