  To any finite local embedding of Deligne--Mumford stacks $g: Y\to X$ we
associate an \'etale, universally closed morphism $F_{Y/X}\to X$ such that for
the complement $Y^2_X$ of the image of the diagonal $Y \to Y\times_XY$, the
stack $F_{Y^2_X/Y}$ admits a canonical closed embedding in $F_{Y/X}$, and
$F_{Y/X}\times_XY$ is a disjoint union of copies of $F_{Y^2_X/Y}$. The stack
$F_{Y/X}$ has a natural functorial presentation, and the morphism $F_{Y/X}\to
X$ commutes with base-change. The image of $Y^2_X$ in $Y$ is the locus of
points where the morphism $Y \to g(Y)$ is not smooth. Thus for many practical
purposes, the morphism $g$ can be replaced in a canonical way by copies of the
closed embedding $F_{Y^2_X/Y}\to F_{Y/X}$.
