Let $F:M \rightarrow N$ be a $C^{\infty}$ map between manifolds. Then we have the pullback $F^*:C^{\infty}(N) \rightarrow C^{\infty}(M)$. From this we can get a map from
$J_{N, F(p)} / J_{N, F(p)}^2 $ to $J_{M, p} / J_{M,p}^2$, where $J_{M,p}$ is the ideal of functions $f$ in $C^{\infty}(M)$ such that $f(p) = 0$. This then gives a map $F^*: T^*_{F(p)}N \rightarrow T^*_p M$ of cotangent spaces (because $T^*_p M$ is isomorphic to $J_{M, p} / J_{M,p}^2$ etc). In the notes I am reading then states:
Dually, there is a map $F_*: T_p M \rightarrow T_{F(p)}N$ of tangent spaces.
I am wondering could someone explain to me how this map $F_*$ is defined? (since there is no explanation in this notes...) Thank you.
PS here $T_pM$ is the collection of linear maps $X : C^{\infty}(M) \rightarrow \mathbb{R} $ such that $$ X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1. $$
Let $X : C^{\infty}(M) \rightarrow \mathbb{R} $ such that $$ X(f_1 f_2) = f_1(p) X f_2 + f_2(p) X f_1,$$ then $F_*:T_pM \rightarrow T_{f(p)}M$ is defined as $(F_*X)(f) = X(f\circ F).$
For more information, I recommend Lee's great book "Introduction to Smooth Manifolds", chapter 3.