Surjective etale maps from schemes give presentations

$R$ is a scheme, $j$ is an equivalence relation, and $s,t$ are etale. By [F1] the fibre product $R=U{\times }_{F}U$ is a scheme and the projections $s,t\text{\hspace{0.17em}⁣}:R\to U$ are the base changes of the representable morphism $f$ along $f$; since $f$ is etale and étale morphisms are stable under base change by [F2], both $s$ and $t$ are etale. The map $j=(t,s)\text{\hspace{0.17em}⁣}:R\to U{\times }_{S}U$ is injective on $T$-points for every scheme $T$, because a $T$-point of $R$ is a pair of $T$-points of $U$ with equal image in $F$, and its image in $U{\times }_{S}U$ is that pair; hence $j$ is a monomorphism. The groupoid operations are the standard kernel-pair operations of the map $f$: the diagonal $U\to R$, the swap $R\to R$, and composition induced by the projections of the triple fibre product; the groupoid axioms hold because they hold for the pair groupoid of $U$ restricted to the subobject of pairs with equal image in $F$. Hence $j$ is an equivalence relation on $U$ over $S$.

The comparison map is a monomorphism. The morphism $f\text{\hspace{0.17em}⁣}:U\to F$ coequalizes $s$ and $t$ by construction of $R=U{\times }_{F}U$, so it induces a morphism of presheaves $P_{U\mathrm{/}R}\to F$ from the naive quotient presheaf, which is injective because two $T$-points of $U$ with equal image in $F$ are by definition a $T$-point of $R$. To see directly that plus preserves this injection, represent two elements of $P_{U\mathrm{/}R}^{+}(T)$ by matching families. If their images in $F^{+}(T)$ agree, the definition of the plus colimit gives a common refining cover on which their images agree. Injectivity of $P_{U\mathrm{/}R}\to F$ makes the original restricted families agree there, so their plus classes coincide. Applying this argument again gives an injection $P_{U\mathrm{/}R}^{++}\to F^{++}$; since $F^{++}\cong F$ by [F4], the induced map $U\mathrm{/}R\to F$ is a monomorphism.

The comparison map is an epimorphism. Let $T$ be a scheme and $\xi \in F(T)$ a section. Since $f$ is representable, etale and surjective, the base change $U{\times }_{F,\xi }T\to T$ is an etale surjective morphism of schemes, so there is an fppf covering $\{T_i\to T\}$ and lifts $T_i\to U$ with $f$-image $\xi {\mathrm{\mid }}_{T_i}$; in other words the section $\xi$ lifts fppf-locally to $U$. The assignment sending a $U$-point to its $f$-image factors through $U\mathrm{/}R$, so every section of $F$ is locally in the image of $U\mathrm{/}R\to F$: the comparison is an epimorphism of sheaves.

Conclusion. For each section of $F(T)$, choose the local preimages supplied by step 2.2. Their restrictions agree on overlaps by the monomorphism of step 2.1, so the sheaf condition on $U\mathrm{/}R$ glues them uniquely to a preimage on $T$. Thus the comparison is bijective on every section set, compatibly with restriction, and $F\cong U\mathrm{/}R$ as fppf sheaves; the diagram $R\rightrightarrows U\to F$ is the coequalizer presenting $F$. Together with steps 1.1 the three assertions hold. The Axiom of Choice is inherited from the quotient-sheaf construction of [F3], which is used in steps 2.1-2.2.