$k$-form $\omega$ is smooth iff it is smooth as a section $\omega \colon M\rightarrow \Lambda ^k(M)$ of $\pi$

Question

Consider $\Lambda ^k(M)=\bigcup_{p\in M}\Lambda ^k(T_{p}M)$ with the natural smooth structure, $M$ a smooth manifold. With this structure on $\Lambda ^k(M)$, the $\pi \colon \Lambda ^k(M)\rightarrow M$ projection is smooth.

Show that a $k$-form $\omega$ is smooth iff it is smooth as a section $\omega \colon M\rightarrow \Lambda ^k(M)$ of $\pi$.

I'm trying to use the definition $\omega(X_1,\dots,X_k)$ is smooth for every collection $X_1,\dots,X_k$ of smooth vector fields to show that $\omega$ is smooth as map but I cannot.

Answer 1

Consider the natural atlas $\{\Lambda ^k(U), (d\varphi^*)^{-1}\}$ for $\Lambda ^k(M)$, where $(U,\varphi)$ chart for $M$ with $\varphi(p)=q$. So,

$(d\varphi^*)^{-1}\circ\omega\circ\varphi(\varphi(p))=(d\varphi^*)^{-1}\circ\omega_{p}\\ =(d\varphi^*)^{-1}_{q}\omega=\omega(d\varphi^{-1}_{q},...,d\varphi^{-1}_{q})$

But are $d\varphi^{-1}_{q}$ fields smooths on M?

I'm not sure what I did is right.