form exact $\Leftrightarrow$ pull-back exact

form exact $\Leftrightarrow$ pull-back exact

Is the form exact $\Leftrightarrow$ pull-back exact? Since $$f^*\omega =
\omega \circ df,$$ which seems irrelavant. Because the composition with
$df$ does not change $\omega$ is exact or not.
The definition I was trying to use:
Suppose $A: V \to W$ is a linear map. Then the transpose map $A^*: W^* \to
V^*$ extends to the exterior algebras, $A^*: \Lambda^p(W^*) \to
\Lambda^p(V^*)$ for all $p>0$. If $T \in \Lambda^p(W^*)$, just define $A^*
T \in \Lambda^p(V^*)$ by $$A^*T(v_1, \dots, v_p) = T(Av_1, \dots, Av_p)$$
for all vectors $v_1, \dots, v_p \in V$. (Page 159)