Given a stack $\mathscr X$ with enough assumptions we obtain a map $\rho: \mathscr X \to X$ to a coarse moduli space. Furthermore, $\rho$ is proper.
I do not understand what it means for $\rho$ to be proper. I originally thought it meant that base change by a scheme yielded a proper map, but I have been told that $\rho$ does not need to be representable (either by algebraic spaces or schemes), so we can't use this definition.
The only other definition that I know is that a map of stacks $f:\mathscr X \to \mathscr Y$ has property $P$ if there exists a chart for $f$ by schemes (which is a diagram such as below) such that $h$ has property $P$. Note in this diagram X and Y are schemes, $Y \to \mathscr Y$ is surjective and smooth, and $\mathscr{X}^\prime$ is the fibre product of the square on the right. This definition is Definition 8.2.6 in Olsson's book Algebraic Spaces and Stacks.
Unfortunately, this definition is only given for properties $P$ which are stable and local on domain (in the smooth topology). But properness is not local. So I am wondering what the definition of properness is? Can we use the same definition in terms of a chart, regardless of whether or not the property is local on the domain?