Hom-stacks and restriction of scalars

Abstract

Fix an algebraic space $S$, and let $\mathcal{X}$ and $\mathcal{Y}$ be separated Artin stacks of finite presentation over $S$ with finite diagonals (over $S$). We define a stack ${\underline{\mathrm{Hom}}}_S(\mathcal{X},\mathcal{Y})$ classifying morphisms between $\mathcal{X}$ and $\mathcal{Y}$. Assume that $\mathcal{X}$ is proper and flat over $S$, and assume fppf locally on $S$ that there exists a finite finitely presented flat cover $Z\to \mathcal{X}$ with $Z$ an algebraic space. Then we show that ${\underline{\mathrm{Hom}}}_S(\mathcal{X},\mathcal{Y})$ is an Artin stack with quasi-compact and separated diagonal