Lemma 39.3.5. Let $j : R \to U \times _ S U$ be a pre-relation. Assume

$s, t$ are unramified,

for any algebraically closed field $k$ over $S$ the map $R(k) \to U(k) \times U(k)$ is an equivalence relation,

there are morphisms $e : U \to R$, $i : R \to R$, $c : R \times _{s, U, t} R \to R$ such that

\[ \xymatrix{ U \ar[r]_ e \ar[d]_\Delta & R \ar[d]_ j & R \ar[d]^ j \ar[r]_ i & R \ar[d]^ j & R \times _{s, U, t} R \ar[d]^{j \times j} \ar[r]_ c & R \ar[d]^ j \\ U \times _ S U \ar[r] & U \times _ S U & U \times _ S U \ar[r]^{flip} & U \times _ S U & U \times _ S U \times _ S U \ar[r]^{\text{pr}_{02}} & U \times _ S U } \]are commutative.

Then $j$ is an equivalence relation.

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