Hom-stacks and restriction of scalars

Abstract

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