Tim wrote:

> We call a function between manifolds smooth if it maps smooth curves to smooth curves...

This is not the usual definition of a smooth function between Fréchet spaces (or Fréchet manifolds); the usual definition is
here.

I believe the definition you're proposing is weaker (except for $\mathbb{R}^n$, where its a theorem in Michor and Kriegl's book that it's equivalent to the usual one). I'm not 100% sure it's weaker! But it's certainly not the one people usually use in the study of Fréchet manifolds.

So, we should fix this or find the mistake I'm making now...