I do not understand what does $\mathbb{U_n}$ symbol mean, is it just any set that we used the $\mathbb{U}$ letter for? So if I understand correctly we want to show that $\mathbb{U_n}$ and $\mathbb{U_a\times U_b}$ is one to one. Why do we already have the correspondence between integers ? Can somebody explain how does the proof work.