TITLE: Representability of morphism of stacks
QUESTION [7 upvotes]: A morphism of Artin stacks $f:X\to Y$ over $\mathbb Q$ is representable by algebraic spaces if and only if its geometric fibres are algebraic spaces. I would like to know if one can use this to prove the following statement.
Let $f:X\to Y$ be a morphism of finite type separated DM stacks over $\mathbb Q$. Suppose that, for any geometric point $x$ of $X$ with $y= f(x)$, the induced morphism on stabilizers $Stab(x)\to Stab(y)$ is injective. Then $f:X\to Y$ is representable by algebraic spaces.

REPLY [4 votes]: This is http://stacks.math.columbia.edu/tag/04Y5 . I quote :
"
lemma
Let $S$ be a scheme contained in $Sch_{fppf}$.
Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of algebraic
stacks over $S$. The following are equivalent:
1 for $U \in Ob((Sch/S)_{fppf})$ the functor $f : \mathcal{X}_U \to\mathcal{Y}_U$ is faithful,
2 the functor $f$ is faithful, and
3 $f$ is representable by algebraic spaces.
"
See also http://stacks.math.columbia.edu/tag/04YY for a fancy reformulation.