Lemma 78.4.4. Let B \to S as in Section 78.3. Let j : R \to U \times _ B U be a pre-relation of algebraic spaces over B. Consider the relation on |U| defined by the rule
If j is a pre-equivalence relation then this is an equivalence relation.
Proof. Suppose that x \sim y and y \sim z. Pick r \in |R| with t(r) = x, s(r) = y and pick r' \in |R| with t(r') = y, s(r') = z. We may pick a field K such that r and r' can be represented by morphisms r, r' : \mathop{\mathrm{Spec}}(K) \to R with s \circ r = t \circ r'. Denote x = t \circ r, y = s \circ r = t \circ r', and z = s \circ r', so x, y, z : \mathop{\mathrm{Spec}}(K) \to U. By construction (x, y) \in j(R(K)) and (y, z) \in j(R(K)). Since j is a pre-equivalence relation we see that also (x, z) \in j(R(K)). This clearly implies that x \sim z.
The proof that \sim is reflexive and symmetric is omitted. \square
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