Let $X_1, X_2$ be vector fields over a smooth manifold $M$ and likewise $Y_1, Y_2$ vector fields over a smooth manifold $N$. Furthermore, let $X_1$ and $Y_1$ aswell as $X_2$ and $Y_2$ be $\Phi$-related for some morphism $\Phi: T M \to T N$ (i.e. we have $\Phi \circ Y_i = X_i \circ \phi$ where $\phi: M \to N$ is the projection mapping of $\Phi$). I now want to show that the commutator/Lie bracket $[X_1, Y_1]$ is also $\Phi$-related to $[X_2, Y_2]$.
Now I tried to go by the definition of the commutator and write out
$$[X_1, Y_1] \circ \phi = X_1 Y_1 \circ \phi - Y_1 X_1 \circ \phi \\ = X_1 \circ (\Phi \circ Y_2) - Y_1 \circ (\Phi \circ X_2) \\$$
But I'm not really sure how to go from there as I don't think I could interchange $X_i \circ \Phi$ for $\Phi \circ X_i$? I would need to show that this expression is equal to
$$\Phi \circ [X_2, Y_2] = \Phi(X_2 Y_2 - Y_2 X_2)$$
but approaching it from this side, I again don't know how to continue from here, as I don't think I could just "pull in" $\Phi$ into the individual summands? Is there something about the commutator that I'm misunderstanding or forgetting? It seems like an easy thing to show yet I'm a bit lost here.