Theorem [efr-0022]

Consider the category of smooth microlinear spaces X equipped with a map X \to A and a vector field X \to TX.

1. This category has a terminal object, given by the colimit taken over a suitable class of small objects. In other words, a point in the terminal system is given by some system U \to TU \times A and a point u_0 \in U, up to an equivalence relation of bisimulation
2. The mapping space A^R, equipped with the evaluation at 0 map A^R \to A and the flow \frac {du}{dt} = \frac {du}{dx} is a subterminal object.
3. A system X \to A \times TX admits a map to A^R if and only if it is integrable in both directions