A Boundedness Theorem for Hom-stacks

The purpose of this note is to prove the following “boundedness” stated in [Ol]. Let X and Y be separated Deligne–Mumford stacks of finite presentation over an algebraic space S and define HomS(X ,Y) as in [Ol, 1.1]. Assume that X is flat and proper over S, and that locally in the fppf topology on S, there exists a finite flat surjection Z → X from an algebraic space Z. Let Y → W be a quasi-finite proper surjection over S to a separated algebraic space W over S of finite presentation. By [Ol, 1.1] we then have Deligne–Mumford stacks HomS(X ,Y) and HomS(X ,W ).