The category $\mathrm{Mfd}$ of finite dimensional manifolds sits fully faithfully inside the larger category $\mathrm{LocProMfd}$ of locally pro-finite dimensional manifolds, which basically extends manifolds by spaces locally modeled on the Fréchet space $\mathbb{R}^\mathbb{N}$. Take a finite dimensional base manifold $\Sigma$ and consider the category $\mathrm{LocProMfd}_{\downarrow \Sigma}$ of fibered locally pro-finite dimensional manifolds over $\Sigma$. The functor $$ J^\infty_\Sigma \colon \mathrm{LocProMfd}_{\downarrow \Sigma} \to \mathrm{LocProMfd}_{\downarrow \Sigma} , \quad (F\to M) \mapsto (J^\infty_\Sigma(F) \to M) , $$ that assigns to a fibered manifold the bundle of jets of its sections over $\Sigma$ is a comonad. The fiber of the infinite order jet bundle of a finite dimensional fibered manifold is already infinite dimensional. So there is no way to escape this enlargement of $\mathrm{Mfd}$, or something like it. This comonadicity observation is due to Michal Marvan, first recorded in a conference proceedings note in 1986 and elaborated in his 1989 PhD thesis at Moscow State University.
Together with Urs Schreiber (arXiv:1701.06238arXiv:1701.06238) we have checked that the jet functor both survives in the much more general setting of synthetic differential geometry and maintains its comonad property for some basic category theoretic reasons (it is a base change comonad of another functor). Check the arXiv preprint for a detailed discussion and precise references to Marvan's original observations.
