Quotient maps of etale equivalence relations are etale surjective

The fibre product is a scheme, fppf-locally on $T$. Let $a\text{\hspace{0.17em}⁣}:T\to F$ and choose the presentation of [F1]. Over $T_i$ the projection $\pi \text{\hspace{0.17em}⁣}:T{\times }_{a,F,c}U\to T$ base changes to ${\pi }_i\text{\hspace{0.17em}⁣}:T_i{\times }_{{\phi }_i,T}(T{\times }_{a,F,c}U)\to T_i$, and by [F3] the source is computed as $T_i{\times }_{a_i,U,t}R$: a $T^{\mathrm{\prime }}$-point is a pair $(t^{\mathrm{\prime }},r)$ with $t^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:T^{\mathrm{\prime }}\to T_i$, $r\text{\hspace{0.17em}⁣}:T^{\mathrm{\prime }}\to R$ and $t(r)=a_i{t}^{\mathrm{\prime }}$, which maps to $U$ by $s(r)$ and thus defines a point of the fibre product. Conversely, equality in $F$ gives a local $R$-witness by [F1]; its uniqueness and represented-sheaf descent make it a unique global witness. Thus this map is an isomorphism of presheaves. Since $t$ is étale, ${\pi }_i$ is the base change of the étale $t$ along $a_i$, hence étale; it is surjective because $t$ is.

$\pi$ is representable, etale and surjective. Representability uses the assumed algebraicity of $F$: the sheaf $T{\times }_{F}U$ is the pullback of ${\mathrm{\Delta }}_F$ along the scheme $T{\times }_{S}U\to F\times F$, hence is a scheme. Each local base change ${\pi }_i$ is etale by step 1.1. Etaleness descends here as follows. Over an affine target open $\mathrm{Spec}A$, the images of affine opens in the fppf cover form an open covering by [F4]. Quasi-compactness selects finitely many covering affine opens; their disjoint union is an affine fppf refinement $\mathrm{Spec}B\to \mathrm{Spec}A$, and $A\to B$ is faithfully flat by [F4]. For an affine source open $\mathrm{Spec}C$, local finite presentation of the base change means $C{\otimes }_{A}B$ is a finitely presented $B$-algebra. Tensor coefficients of its finitely many generators supply finitely many generators of $C$ over $A$, by faithful flat detection of the quotient module. For the resulting presentation $A[x_1,\dots ,x_m]\twoheadrightarrow C$, its kernel extends to the kernel over $B$ by flatness; finitely many tensor coefficients of generators of this extended ideal generate the original ideal by faithful flat detection. Thus $C$ is finitely presented. Flatness descends by [F2], and ${\mathrm{\Omega }}_{C\mathrm{/}A}$ vanishes because its scalar extension vanishes by differential base change and faithful flat detection; [F2] then gives etaleness. Surjectivity descends on points, since the cover is onto and each ${\pi }_i$ is onto. Therefore $c$ is representable, etale and surjective.

The presentation. By step 2.1 the morphism $c$ is representable, etale and surjective, and $R=U{\times }_{F}U$ is the kernel pair of $c$ by the local-witness and descent argument of step 1.1; hence $(U,R,U\to F)$ is a presentation of the algebraic space $F$ in the sense of Presentations of algebraic spaces. The Axiom of Choice is inherited from the quotient-sheaf construction of [F1].