If you have that \(\displaystyle \left\{f_\alpha\right\}_{\alpha\in\mathcal{A}}\) is a set of bijections from the corresponding \(\displaystyle \left\{U_\alpha\right\}_{\alpha\in\mathcal{A}}\) to the corresponding \(\displaystyle \left\{V_\alpha\right\}_{\alpha\in\mathcal{A}}\) then \(\displaystyle \prod_{\alpha\in\mathcal{A}}f_\alpha:\prod_{\alpha\in\mathcal{A}}U_\alpha\to\prod_{\alpha\in\mathcal{A}}V_\alpha:\prod_{\alpha\in\mathcal{A}}\{x_\alpha\}\mapsto\prod_{\alpha\in\mathcal{A}}\left\{f_\alpha(x_\alpha)\right\}\) is a bijection. Or, for just the two case that

**MoeBlee** stated just take \(\displaystyle f_1\times f_2:U_1\times U_2\to V_1\times V_2

x_1,x_2)\mapsto (f_1(x_1),f_2(x_2))\)

Prove it!