Given a smooth map $f: M\rightarrow N$ between smooth manifolds, by definition (wiki reference or this post) we have a pull back tangent bundle $f^{*}TN\subset M\times TN$.
Can we define a Lie bracket $[X,Y]$ for $X,Y\in f^{*}TN$?
(It should because I have seen some property like $f_{*}([X,Y])=[f_{*}X,f_{*}Y]$ where $X,Y\in TM$. (ref: Answers in this post))
But there's a problem, in convention $[X,Y]=XY-YX$, but what does $X(Y)-Y(X)$ mean exactly for $X, Y\in f^{*}TN$? For $p\in M$, given $h\in C^{\infty}{(V)}$, where $f(p)\in V$ open in $N$, it seems we should have $[X,Y]|_{p} h =X_p({Y(h)})-Y_p({X(h)})$. But $X_p\in T_{f(p)}N$, therefore $Y(h)$ should be in $C^{\infty}(V)$, but how could it be defined?