Quotients of schemes by etale equivalence relations are algebraic spaces

Reduction to a disjoint union of affines. Let $U^{\mathrm{\prime }}={\coprod }_i{U}_i\to U$ be the disjoint union of the members of an affine open covering of $U$. The family is a surjective étale morphism, hence flat and locally of finite presentation; by [F1] the restriction $R^{\mathrm{\prime }}$ of $R$ to $U^{\mathrm{\prime }}$ is an étale equivalence relation, and by [F2] applied to the jointly surjective morphism $U^{\mathrm{\prime }}\to U$ the induced map $U^{\mathrm{\prime }}\mathrm{/}{R}^{\mathrm{\prime }}\to U\mathrm{/}R$ is an isomorphism. Hence we may replace $U$ by the disjoint union of affine schemes $U_i$.

The affine pieces. Let $R_i$ be the restriction of $R$ to $U_i$; by [F1] it is an etale equivalence relation, and by [F3] the quotient $F_i=U_i\mathrm{/}{R}_i$ is an algebraic space with $U_i\to F_i$ representable, etale and surjective. The canonical morphisms $F_i\to F=U\mathrm{/}R$ are representable open immersions by [F2], and the induced map ${\coprod }_i{F}_i\to F$ is surjective as a morphism of sheaves because the $U_i$ cover $U$ and $F$ is the quotient sheaf of $U$: every section of $F$ lifts fppf-locally to $U$, hence to some $U_i$.

The coproduct is an algebraic space. The morphism ${\coprod }_i{U}_i\to {\coprod }_i{F}_i$ is a disjoint union of the representable etale surjective covers $U_i\to F_i$, hence representable, etale and surjective, and its source ${\coprod }_i{U}_i$ is a scheme; by clause (1) of [F4] the coproduct ${\coprod }_i{F}_i$ is an algebraic space.

Gluing. The hypotheses of clause (2) of [F4] are satisfied: $F$ is an fppf sheaf, each $F_i\to F$ is representable and an open immersion, the map ${\coprod }_i{F}_i\to F$ is surjective, and ${\coprod }_i{F}_i$ is an algebraic space by step 1.3. Hence $F=U\mathrm{/}R$ is an algebraic space over $S$, and $U\to F$ is representable, etale and surjective by [F5], so $(U,R,U\to F)$ is a presentation. The Axiom of Choice is inherited from the quotient and descent suppliers used in [F2]-[F3].