One-to-one correspondense between proper families of Boolean functions and unique sink orientations of cubes

In the paper, we study the relationship between proper families of Boolean functions and unique sink orientations of Boolean cubes. A family of Boolean functions $ F = (f_1(x_1, \ldots, x_n), \ldots, f_n(x_1, \ldots, x_n))$ is called proper if for every two binary vectors $\alpha, \beta$, $\alpha \ne \beta$, the following condition holds:
$$ \exists i (\alpha_i \ne \beta_i\ \&\ f_i(\alpha) = f_i(\beta)).$$
Unique sink orientation of Boolean cube $\mathbb{E}_n$ is such an orientation of edges of $\mathbb{E}_n$ that any subcube of $\mathbb{E}_n$ has a unique sink, i.e., a unique vertex without outgoing edges. The existence of one-to-one correspondence between two classes of objects is proved, and various properties are derived for proper families. The following boundary for the number $T(n)$ of proper families of given size $n$ is obtained: there exist two numbers $B$ and $A$, $B \ge A > 0$, such that $n^{A 2^n} \le T(n) \le n^{B 2^n}$ for $n \ge 2$. Also, coNP-completeness of the problem of recognizing properness is derived.