I know that for a given coordinate system of the tangent space $\partial_1,..,\partial_n$ the coordinate map is smooth, as the coordinate map $\pi_i$ of ${T_xM}$ is nothing but the composition of the chart and coordinate map in $\mathbb{R}^n$ and both of them are smooth. Unfortunately, this is less clear to me if we are talking about the cotangent space, cause then I have a coordinate system $dx_1,..,dx_n$ , but how do I see that now the coordinate projection is still smooth?
If anything is unclear, please let me know.