Lemma 83.3.5. In the situation of Lemma 83.3.3 there is an isomorphism of sheaves
\[ U'/R' = X' \times _ X U/R \]
For the construction of quotient sheaves, see Groupoids in Spaces, Section 78.19.
Proof. Since $U \to X$ is $R$-invariant, it is clear that the map $U \to X$ factors through the quotient sheaf $U/R$. Recall that by definition
\[ \xymatrix{ R \ar@<1ex>[r] \ar@<-1ex>[r] & U \ar[r] & U/R } \]
is a coequalizer diagram in the category $\mathop{\mathit{Sh}}\nolimits $ of sheaves of sets on $(\mathit{Sch}/S)_{fppf}$. In fact, this is a coequalizer diagram in the comma category $\mathop{\mathit{Sh}}\nolimits /X$. Since the base change functor $X' \times _ X - : \mathop{\mathit{Sh}}\nolimits /X \to \mathop{\mathit{Sh}}\nolimits /X'$ is exact (true in any topos), we conclude. $\square$
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