Does a bijection exists between $U$ and $W$?

Question

I have the following bijections:

$$Ψ:U→V$$

$$Ψ⁻¹:W→S$$

$$ξ₁:U→S$$

$$ξ₂:V→W$$

there is no known relations between $U$ and $W$ and $V$ and $S$.

My question is: Does a bijection exists between $U$ and $W$?

Answer 1

YES. Let $f = \xi_2(\Psi )$. It is the bijection from $U$ to $W$.