f-related vector fields

Let $M$ and $N$ be manifolds and let $f : M \to N$ be a smooth map. A vector field $X$ on $M$ and a vector field $Y$ on $N$ are said to be $f$ -related if

d f_{p} (X_{p}) = Y_{f (p)}, \forall p \in M .

This means that for every function

h : N \to R

X_{p} [h \circ f] = Y_{f (p)} h .

Suppose that $X$ is $f$ -related to $Z$ and $Y$ is $f$ -related to $W$ . Show that $[X, Y]$ is $f$ -related to $[Z, W]$ .