I am going through Nakahara's textbook on geometry and topology in physics. Intuitively, I understand the definition of a lie derivative of a vector field
$\mathcal{L}_xY = \lim_{\epsilon_\to 0}\frac{1}{\epsilon}[(\sigma_{-\epsilon})_*Y|_{\sigma_\epsilon(x)} - Y|_x]$
We use the above definition as we cannot simply take the difference of the two vectors as they are on different tangent spaces. So we map one of the vectors to the tangent space of the other using the induced map $(\sigma_{-\epsilon})$.
My problem is in understanding this definition for one-forms. The equation given is
$\mathcal{L}_x\omega = \lim_{\epsilon_\to 0}\frac{1}{\epsilon}[(\sigma_{\epsilon})^*\omega|_{\sigma_\epsilon(x)} - \omega|_x]$
Here, the induced map on $\omega$ seems to be along $\epsilon$. I do not understand how that brings it into the tangent space at $x$. I think my understanding of the pullback used here is lacking.