Dynamical Systems as a Functor Category

According to this blog post by [Jade](https://jadeedenstarmaster.wordpress.com/2019/03/31/dynamical-systems-with-category-theory-yes/) we can represent dynamical systems as a functor category from the reals to differentiable manifolds where the objects are ϕ: ℝ→Diff and the morphisms are natural transformations. Does this mean we could build a GAT for dynamical systems? Could we then take a presentation of a subcategory where we can express the natural transformations algebraically and then generate big dynamical systems compositionally?