Definition of pull-back
Let $M$ and $N$ be two manifolds. The pull back of $f:M\to N$ is defined as
$$[(f^*\omega)(x)](\omega_1,\ldots,\omega_r)=\omega(f(x))\cdot(f'(x)\cdot\omega_1,\ldots,f'(x)\omega_r)$$
Where $\omega$ is a differential form of degree $r$ in $N$, $x\in M$ and $\omega_1,\ldots,\omega_r\in T_xM$.
Thus it takes a form $\omega\in N$ to a form $f^*\omega\in M$.
What I didn't understand
Let's take now a differential form $\omega$ of degree $r$ and class $C^1$ in $M$, i.e., $\omega$ is a function $x\in M\mapsto\omega(x)\in A_r(T_xM)$. The book I'm reading says for every parametrization $\varphi:U_0\to U$ in $M$, there is a unique form $d_{\varphi}\omega$ of degree $r+1$ in $U$ such that $\varphi^*(d_{\varphi}\omega)=d(\varphi^*\omega)$. (afterwards he defines this as being the exterior derivative $d\omega$).
So what is $\varphi^*$? the author of the book didn't define the pull-back of a parametrization. Is there some abuse of notation here?
I've been passing hours to try to understand this, I really need help to get this definition.
Remark: if you don't know/understand my notation, please leave a comment.