I am reading Guillemin and Pollack's Differential Topology Page 163:
If $f: X \to Y$ is a smooth map and $\omega$ is a $p$-form on $Y$, define a $p$-form $f^*\omega$ on $X$ as follows: $$f^*\omega(x) = (df_x)^*\omega[f(x)].$$
So my question is, what is $\omega[f(x)]$?
I have been trying to make this question self-contained, but here is the whole background: Definition of pullback. In that question, I have shown $(df_x)^*T(v_1, \dots, v_p) = T \circ df_x (v_1, \dots, v_p)$ and $\omega$ is $(df_x)^*T$. But I want the missing piece about what that $\omega[f(x)]$ is equal to, to make sense of the definition of $f^* \omega (x)$.