To any finite local embedding of Deligne--Mumford stacks g:Y→X we associate an étale, universally closed morphism FY/X→X such that for the complement Y2X of the image of the diagonal Y→Y×XY, the stack FY2X/Y admits a canonical closed embedding in FY/X, and FY/X×XY is a disjoint union of copies of FY2X/Y. The stack FY/X has a natural functorial presentation, and the morphism FY/X→X commutes with base-change. The image of Y2X in Y is the locus of points where the morphism Y→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 FY2X/Y→FY/X