Does $\varphi$-relation of vector fields respect the Lie bracket?

Question

Let $M$ and $N$ be two smooth manifolds and $\varphi : M \longrightarrow N$ be a smooth map. Let $X,Y \in \mathfrak {X} (M)$ and $Z,W \in \mathfrak {X} (N)$ be smooth vector fields on $M$ and $N$ respectively such that $X \sim_{\varphi} Z$ and $Y \sim_{\varphi} W$ i.e. for any $p \in M$ we have $(d_p \varphi) (X_p) = Z_{\varphi (p)}$ and $(d_p \varphi) (Y_p) = W_{\varphi (p)}.$ Then is it true that the same holds for their Lie brackets i.e. $[X,Y] \sim_{\varphi} [Z,W]\ $?