Right adjoint to forgetful functor from "dynamical system" digraph Question about "dynamical systems," as Lawvere/Schnauel calls them in their baby book (ie digraph w exactly 1 arrow out of each point). What would a "chaotic" dynamical system be? In the book's second-edition appendices, they say the forgetful functor to Set has a right adjoint, associated with such a system, but I can't envision what such an object would be--what's "indiscreteness" in a deterministic graph? ...On that note, by the way, they also say that the left adjoint assigns the "free dynamical system" and say it has to do with natural numbers and recursion, but as far as I can see the left adjoint to the forgetful functor to set could only be the discrete functor, same as for any digraph.
 A: Let's be precise here. The category $\text{Dyn}$ of dynamical systems is the category whose objects are pairs $(X, f)$ where $X$ is a set and $f : X \to X$ is an endomorphism, and whose morphisms are maps of underlying sets $\phi : X_1 \to X_2$ such that $f_2 \circ \phi = \phi \circ f_1$. $\text{Dyn}$ has a forgetful functor $U : \text{Dyn} \to \text{Set}$ sending $(X, f)$ to $X$. The claim is that $U$ has both a left and a right adjoint.
The left adjoint sends a set $X$ to the dynamical system $(X \times \mathbb{N}, f)$ where $f(x, n) = (x, n+1)$ (exercise). This is what Lawvere and Schanuel mean by the "free dynamical system" on $X$.
The right adjoint sends a set $X$ to the dynamical system $(X^{\mathbb{N}}, f)$ of sequences $x(n)$ of elements of $X$, where $f(x(n)) = x(n+1)$ is the left shift (exercise). This is the "cofree dynamical system" on $X$. 
You shouldn't be thinking of dynamical systems as graphs for the purposes of doing either of these exercises. There is a functor embedding $\text{Dyn}$ into the category of digraphs, but you have no guarantee that this embedding respects adjoints in any sense. Think, for example, of the embedding of abelian groups into groups.
