"Let $G_1. G_2$ be groups. Prove that the map
$$ \varphi: G_1 \times G_2 \rightarrow G_2 \times G_1$$ $$ \varphi: (g_1, g_2) \mapsto (g_2, g_1)$$
defines an isomorphism between $G_1 \times G_2$ and $G_2 \times G_1$.
I can prove the homomorphism bit but in the answers, it just says "Obviously $\varphi$ is bijective". Why is this obvious? Is it because it is commutative?