Let $\mathbb{P}_3 = (\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1)/S_3$ be $3$-symmetric product of Riemann sphere $\mathbb{P}^1$, show that $\mathbb{P}_3 \simeq \mathbb{P}^3$ is holomorphic isomorphism.
Can we define $g: \mathbb{P}_3\to \mathbb{P}^3$ as $$ g: ([x_1:y_1],[x_2:y_2],[x_3:y_3]) \mapsto [x_1x_2x_3+x_1x_2y_3:x_1y_2x_3+x_1y_2y_3+y_1x_2x_3+y_1x_2y_3:y_1y_2x_3+y_1y_2y_3:x_1x_2x_3:y_1y_2y_3] $$ ? And how to prove it is a holomorphic isomorphism? Thanks.