In Hamilton's paper on the Nash-Moser inverse function theorem he shows that if $M$ is a smooth compact manifold and $V\to M$ a smooth vector bundle then its smooth sections $\Gamma(V)$ equipped with the norms $||\sigma||_n:= \sum_{j=1}^n|\nabla^j\sigma|$ for $\sigma\in \Gamma(V)$ is a (tame) Fréchet space. Similarly he shows that $\mathrm{Diff}(M)$ the diffeomorphism group of $M$ is a tame Fréchet Lie group.

My question is the following. We regard the pullback of differential forms as a map $$\mathrm{Diff}(M)\times \Omega^n(M)\to \Omega^n(M)$$ $$(f,w)\mapsto f^*w$$ of Fréchet manifolds and Fréchet spaces. Here $\Omega^n(M)$ denotes the Fréchet space with the topology of smooth sections of a vector bundle as illustrated above.

My question is whether this map was shown somewhere to be smooth tame.

I believe it should be similar to Hamilton's proof (in the same paper) that nonlinear differential operators are smooth tame the only difference being that $(f^*w)_p$ depends on $w_{f(p)}$ and not $w_p$. And since it suffices to look at diffeomorphisms close to the identity the proof should still go through. Any literature recommendations on this subject are greatly appreciated. Thank you!

(p.s. This is my first question on here so you are more than welcome to point out any issues with my question. Thank you!)