Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$.

Assume there is a quasifinite, representable, surjective morphism $f:X\rightarrow\ Y$.

Is there a canonical way to write $f$ as $X\rightarrow Z \rightarrow Y$ where $X\rightarrow Z$ is an open immersion and $Z\rightarrow Y$ is finite representable? That is to describe $Z$ in terms of $X,Y$.

I'm not sure $Spec(C)$, where $C$ is the integral closure of $\mathcal O_Y$ in $f_{*} \mathcal O_X$ works as in Laumon-Moret Bailey, or does it? Also, I'm not assuming $Y$ is normal so the Stein factorization might not work.

Let $X$ be a separated Deligne Mumford stack of finite type over a base field $k$ and $Y$ be a proper Deligne Mumford stack over $k$.

Assume there is a quasifinite, representable, surjective and smooth morphism $f:X\rightarrow\ Y$.

Is there a canonical way to write $f$ as $X\rightarrow Z \rightarrow Y$ where $X\rightarrow Z$ is an open immersion and $Z\rightarrow Y$ is finite representable? That is to describe $Z$ in terms of $X,Y$.

I'm not sure $Spec(C)$, where $C$ is the integral closure of $\mathcal O_Y$ in $f_{*} \mathcal O_X$ works as in Laumon-Moret Bailey, or does it? Also, I'm not assuming $Y$ is normal so the Stein factorization might not work.