I'm a little confused about Lemma 9.7 of Tu's An Introduction to Manifolds. Below is the statement of the lemma and the proof.
I don't really understand how Tu is saying that $f_{\ast,p}$ equals $g_{\ast,p}$ for all $p \in N$. These maps have different codomains $T_{f(p)}\mathbb{R}$ and $T_{g(p)}\mathbb{R}$ (as vector spaces they are isomorphic however) respectively but the derivations depend on the bases of these spaces. If anyone could explain what Tu means by $f_{\ast,p}$ equals $g_{\ast,p}$ that would be great.