This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata.

Let $X$ be a normal quasiprojective variety such that the canonical divisor $K_X$ is a $\mathbb{Q}-$Cartier divisor. Each point $x \in X$ has an open neighborhood $U_x$ such that $m_xK_X$ is a principal Cartier divisor on $U_x$ for a minimum positive integer $m_x$. The canonical covering $\pi_x: \tilde U_x \to U_x$ is a finite morphism of degree $m_x$ from a normal variety which is etale in codimension $1$ and such that $K_{\tilde U_x}$ is a Cartier divisor. The canonical coverings are etale locally uniquely determined, thus we can define the canonical covering stack $\mathcal{X}$ as the stack above $X$ given by the collection of canonical coverings $\pi_x: \tilde U_x \to U_x$.

I have two question about this construction:

(1) Why "The canonical coverings are etale locally uniquely determined"?

(2)Should I think the canonical covering stack $\mathcal{X}$ as a gluing of stacks $Sch / U_x$ or as a gluing of quotient stack $[U_x / \mu_x]$(I only vaguely thought $\mu_x$ should be some group in the construction in the etale covering).