Example Clock system [efr-9WQU]

Consider the theory of discrete-time, discrete dynamical systems from Example [efr-002F]. Let \xi : TS \leftrightarrows {I \choose O} be a system. Consider the system c: T\mathbb {N} \leftrightarrows {* \choose \mathbb {N}} given by n \mapsto n in the forwards direction, and (n,*) \mapsto n+1 in the backwards direction. (To be clear, the object denoted {* \choose \mathbb {N}} is the map \mathbb {N} \to \mathbb {N}---it is a singleton in each fiber). Then a morphism of systems c \to \xi consists of the following data:

1. A function x: \mathbb {N} \to S.
2. Another function o: \mathbb {N} \to O.
3. A third function i: \mathbb {N} \to I so that i(n) \in I_{o(n)}.
4. So that o(n) = \xi (x(n)) and \xi ^\#(x(n),i(n)) = x(n+1)

That is, it is a choice of a sequence x_n of points in the state-space, and a sequence of inputs i_n compatible with the outputs, so that this sequence obeys the dynamics. In other words, it is a trajectory of the system.