Natural numbers can be defined as the initial object of the category of pointed dynamical systems with triple $\left(X, s_0, f\right)$ where $f:X \rightarrow X$ and $s_o \in X$, as objects and conjugacy of dynamical systems as morphisms, i.e. a morphism $$\alpha: \left(X, s_0, f\right) \longrightarrow \left(Y, t_0, g\right)$$ satisfy $\alpha\circ f =g \circ \alpha$ and $\alpha \left(s_0\right)=t_0$.
Is it possible to enrich the following category, in order to be able to define the real numbers, as an initial object? I will be equally content to see how we can define computable numbers as some initial object. Many thanks.