Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?


This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.



Image of a projective variety is closed
To elaborate on Dori's comment, consider $[\mathbf{A}^1/\mathbf{G}\_m]$ which consists of two points: The closed point corresponding to the origin and the orbit of $1$ (which is open). Take $\operatorname{Spec} k \to [\mathbf{A}^1/\mathbf{G}\_m]$ corresponding to this open point. The image is not closed and certainly $[\mathbf{A}^1/\mathbf{G}\_m]$ is not isomorphic to $\operatorname{Spec} \Gamma(\mathbf{A}^1, \mathcal{O}\_{\mathbf{A}^1})^{\mathbf{G}\_m} = \operatorname{Spec} k$, for instance because the stack has two points.


