Affine finite locally free equivalence relations have finite locally free scheme quotients

The equivalence-relation hypothesis includes that $j$ is a monomorphism. Reflexivity supplies the diagonal arrow $e:U\to R$ with $j\circ e={\mathrm{\Delta }}_{U\mathrm{/}k}$, symmetry supplies the unique arrow $i:R\to R$ with $j\circ i=\sigma \circ j$ for the factor swap $\sigma$, and transitivity supplies the unique composition arrow through $j$; these are exactly the groupoid-scheme arrows dual to the identities of the relation, so $(R\rightrightarrows U)$ is an equivalence-relation groupoid with finite locally free source and target.

Applying [F1] to the groupoid produced in step 1.1 gives that $C=\{a\in A:s^{\mathrm{♯}}(a)=t^{\mathrm{♯}}(a)\}$ is a finite-type $k$-algebra, $U\to M=\mathrm{Spec}C$ is finite locally free and surjective, and $R\to U{\times }_{M}U$ is an isomorphism; all hypotheses coincide because both statements use the comorphisms $s^{\mathrm{♯}},t^{\mathrm{♯}}$ of $s,t$ on coordinate rings.

The representing claim is a claim about the same object: by [F2] the phrase "$M$ represents the fppf quotient sheaf $U\mathrm{/}R$" means that $h_M$ is naturally isomorphic to the sheafification of the naive quotient presheaf of $s,t:R\to U$ in the fppf topology, which is exactly the conclusion recorded here; no further hypothesis is added and no step of the published proof is repeated.