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Let $X$ be an $S$-scheme and $G$ a finite group acting on $X$ over $S$. Denote the quotient stack $[X/G]$ by $\mathcal{X}$ and consider a schematic morphism of stacks $\mathcal{Y} \to \mathcal{X}$. Then the fiber product $X \times_{\mathcal{X}} \mathcal{Y}$ is representable by a scheme, say $Y$, on which $G$ acts in such a way that the first projection $Y \to X$ is $G$-equivariant.

My question is: under which conditions (if any) on the morphism $\mathcal{Y} \to \mathcal{X}$ is it true that the second projection $Y \to \mathcal{Y}$ induces an isomorphism between $[Y/G]$ and $\mathcal{Y}$?

P.S. I hope this is not a "stupid" question, but I'm not comfortable with stacks, so any help would be appreciated.

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    $\begingroup$ No conditions are required. Basically, you have to check that $Y\to\mathcal{Y}$ is a presentation of the stack $\mathcal{Y}$, and that $Y\times_{\mathcal{Y}}Y$ is identified with $Y\times G$; both claims follow by base change from the corresponding claims about $\mathcal{X}$. (BTW, technically, if $\mathcal{Y}\to\mathcal{X}$ is representable, it may happen that $Y$ is an algebraic space, not a scheme, but this does not affect anything.) $\endgroup$
    – t3suji
    Feb 29, 2016 at 14:54
  • $\begingroup$ Thank you! I wanted to say "schematic", I will edit this. $\endgroup$
    – user85435
    Feb 29, 2016 at 14:59

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