The Stacks project

Lemma 101.3.2. Let $\mathcal{X}$ be an algebraic stack. Let $T$ be a scheme and let $x, y$ be objects of the fibre category of $\mathcal{X}$ over $T$. Then

  1. $\mathit{Isom}_\mathcal {X}(y, y)$ is a group algebraic space over $T$, and

  2. $\mathit{Isom}_\mathcal {X}(x, y)$ is a pseudo torsor for $\mathit{Isom}_\mathcal {X}(y, y)$ over $T$.

Proof. See Groupoids in Spaces, Definitions 78.5.1 and 78.9.1. The lemma follows immediately from the fact that $\mathcal{X}$ is a stack in groupoids. $\square$

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