The Stacks project

Lemma 77.22.1. Assume $B \to S$, $(U, R, s, t, c)$ and $\pi : \mathcal{S}_ U \to [U/R]$ are as in Lemma 77.20.2. Let $S'$ be a scheme over $S$. Let $x, y \in \mathop{\mathrm{Ob}}\nolimits ([U/R]_{S'})$ be objects of the quotient stack over $S'$. If $x = \pi (x')$ and $y = \pi (y')$ for some morphisms $x', y' : S' \to U$, then

\[ \mathit{Isom}(x, y) = S' \times _{(y', x'), U \times _ S U} R \]

as sheaves over $S'$.

Proof. Let $[U/_{\! p}R]$ be the category fibred in groupoids associated to the presheaf in groupoids ( as in the proof of Lemma 77.20.2. By construction the sheaf $\mathit{Isom}(x, y)$ is the sheaf associated to the presheaf $\mathit{Isom}(x', y')$. On the other hand, by definition of morphisms in $[U/_{\! p}R]$ we have

\[ \mathit{Isom}(x', y') = S' \times _{(y', x'), U \times _ S U} R \]

and the right hand side is an algebraic space, therefore a sheaf. $\square$

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