  Given a separated and locally finitely-presented Deligne-Mumford stack $\cX$
over an algebraic space $S$, and a locally finitely-presented
$\OO_{\cX}$-module $\cF$, we prove that the Quot functor
$\text{Quot}(\cF/\cX/S)$ is represented by a separated and locally
finitely-presented algebraic space over $S$. Under additional hypotheses, we
prove that the connected components of $\text{Quot}(\cF/\cX/S)$ are
quasi-projective over $S$.

Algebraic Geometry