I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89.
From reading elsewhere online it seems convention is to define the induced map of the pushforward of a differentiable function $f: \mathbb R^n \to \mathbb R^m$ with the corresponding linear transformation $Df(p): \mathbb R^n \to \mathbb R^m$ as $$f_*: \mathbb R^n_p \to\mathbb R^m_{f(p)}$$$$f_*(v_p) = (Df(p)(v))_{f(p)}$$ and to then use this definition for the pullback, defined as $$f^*:\Lambda(\mathbb R^m_{f(p)})\to \Lambda(\mathbb R^n_p)$$$$f^*\omega(p)(v_1, .., v_k) = \omega(f(p))(f_*(v_1),..., f_*(v_k)),$$where $\omega$ is a k-form on $\mathbb R^m.$
However Spivak has offered the induced definition for the pullback as $$(f^*\omega)(p) = f^*(\omega(f(p))).$$ which then leads to the above definition.
I'd be grateful for any help explaining the intuition behind this.