Let $\mathcal{S}$ be an appropriate category of smooth spaces, and let $\mathrm{Vect} \colon \mathcal{S}^\mathrm{op} \to \mathsf{Cat}$ be the functor sending a space to the category of vector bundles over it, with action on vector bundles via pullback.
Similarly, let $\mathrm{LieAlg} \colon \mathcal{S}^\mathrm{op} \to \mathsf{Cat}$ be the functor sending a space to the category of Lie algebroids over it.
We can take the category of Grothendieck lenses for each of these functors, and there is a forgetful functor $\mathsf{Lens}(\mathrm{LieAlg}) \to \mathsf{Lens}(\mathrm{Vect})$. Does this functor have a right adjoint? This would be something like the "cofree" Lie algebroid on a vector bundle.
So, specifically, for a vector bundle $E \choose B$, for any "dynamical system" on $E \choose B$, i.e. a choice of $TX \choose X$ with a lens to $E \choose B$, there would be a unique map of Lie algebroids $TX \choose X$ to the cofree Lie algebroid on $E \choose B$. So the cofree Lie algebroid is something like the "universal dynamical system on interface $E \choose B$" -- it can simulate any other dynamical system if you start it in the right state.
The intuition for why this might be the case is that if we work discretely in $\mathsf{Poly}$ (the category of Grothendieck lenses for the slice category functor $\mathsf{Set}^\mathrm{op} \to \mathsf{Cat}$), then I think the analogous category to $\mathsf{Lens}(\mathrm{LieAlg})$ is the category of polynomial comonoids (see Chapter 7 of the poly book). And this case, we do have an analogous construction; the cofree comonoid $\mathfrak{C}_p$ is kind of like the "universal discrete dynamical system" on the interface $p$.
Of course, this is not going to work in the category of finite-dimensional manifolds, but if we could get it to work in the category of microlinear spaces in a smooth topos, I would be happy.
