Let $f : \mathcal{X}\rightarrow\mathcal{Y}$ be a morphism of algebraic stacks which is a gerbe in the sense of the stacks project. Must $f$ induce a surjection of etale fundamental groups? I'm happy to assume that $\mathcal{Y}$ admits a coarse space $Y$.
By Theorem 7.11 in Noohi, the coarse map $\mathcal{Y}\rightarrow Y$ induces a surjection on fundamental groups, so for my question it would suffice to show that the composition $\mathcal{X}\rightarrow\mathcal{Y}\rightarrow Y$ is also a coarse space for $\mathcal{X}$. Is this true? Gerbes are universal homeomorphisms, and at least if we work with the etale topology then gerbes induce bijections on sets of $k$-points for algebraically closed fields $k$, but it's not clear to me why the universal property of coarse spaces must hold.
If not, are there any reasonable positive results?
