Let $S_2$ be an orientable surface and let $$ \varphi : S_2 \to S_1 $$ be a diffeomorphism. Prove that $S_1$ is orientable.
My attempt:
As $S_2$ is orientable, then exists a differentiable field of normal unit vectors $N : S_2 \to \mathbb{R}^3$ over $S_2$. Moreover, $N\circ \varphi : S_1 \to \mathbb{R}^3$ is a differentiable field of normal unit vectors over $S_1$ because $\varphi$ is a diffeomorphism. This implies that $S_1$ is orientable.
What do you think about my proof?