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Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?

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Because

first, $Y\to [Y/G]$ is a chart

second, the following diagram

$$\require{AMScd} \begin{CD} X @>>> Y \\ @VVV @VVV \\ [X/G] @>>> [Y/G] \end{CD} $$ is cartesian (this is so because any morphism between two $G$-torsors is an isomorphism).

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