I am studying article On the Lagrange-Dirichlet converse in dimension three, by Burgos. I am trying to understand this Lemma 2.13, more specifically, understanding the lemma and where it came from, as I've never heard of pullback in mechanics, ode or calculus materials. And the materials on pullback that I found are in the area of category theory. I also accept recommendations for materials that help me understand this lemma.
Lemma 2.13 (Pullback). Let V be a smooth vector field in M tangent to C. Then, there is a unique smooth vector field $V^{*}$ in $Bl_{C}(M)$ tangent to $π^{−1}(C)$ such that:
(1) It satisfies $d_aπ({V_a}^*) = V_{π(a)}$ for every a in $Bl_{C}(M)$.
(2) Restricted to $π^{−1}(C)$, it is the sum of the horizontal lifting of $V_C$ and a vertical vector field.