Flat locally finitely presented restrictions give open subquotients

The saturated open. Put $W^1=g(U^{\mathrm{\prime }})\subseteq U$, open because $g$ is universally open by [F3], and $W=t(s^{-1}(W^1))\subseteq U$, open because $t$ is universally open by [F3]. The set $W$ is saturated: if $u=t(r)$ with $s(r)\in W$, say $s(r)=t(r^{\mathrm{\prime }\mathrm{\prime }})$ with $s(r^{\mathrm{\prime }\mathrm{\prime }})\in W^1$, then lift $r,r^{\mathrm{\prime }\mathrm{\prime }}$ to a common residue-field extension over their shared point (the tensor product of their residue fields is nonzero); transitivity gives $r^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}$ with $t(r^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }})=u$ and $s(r^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }})=s(r^{\mathrm{\prime }\mathrm{\prime }})$, so $u\in W$; the reverse inclusion is immediate from the identity of $U$. Hence $t(s^{-1}(W))=W$, and $W$ is the saturated open generated by $W^1$.

Injectivity and image of $F^{\mathrm{\prime }}\to F$. The morphism $U^{\mathrm{\prime }}\to U$ induces a map of quotient sheaves $F^{\mathrm{\prime }}\to F$. If $x,y\in U^{\mathrm{\prime }}(T)$ have the same image in $F(T)$, then after an fppf covering of $T$ there is a witness $r\in R$ with $t(r)=g(x)$, $s(r)=g(y)$; both $g(x)$ and $g(y)$ then lie in $W^1$, so the witness lands in $g(U^{\mathrm{\prime }})$ in both coordinates and defines a point of $R^{\mathrm{\prime }}$, whence $x,y$ already have the same image in $F^{\mathrm{\prime }}(T)$. Thus $F^{\mathrm{\prime }}\to F$ is injective. Moreover a section $u\in U(T)$ lies in the image of $F^{\mathrm{\prime }}$ exactly when, fppf-locally, it is $R$-equivalent to a point of $W^1$, that is, exactly when $u$ factors through the open $W$. For the reverse implication, the map $R{\times }_{s,U}{U}^{\mathrm{\prime }}\to W$ induced by $t$ is surjective flat and locally finitely presented: it is a composition of base changes of $g$ and $t$, and its image is $W$. Pulling it back along $u:T\to W$ supplies an fppf cover with the desired representative in $U^{\mathrm{\prime }}$. For arbitrary sections of $F^{\mathrm{\prime }}$ injectivity follows by choosing local representatives in $U^{\mathrm{\prime }}$ and applying the sheaf uniqueness condition.

Local presentation of a section. Let $a\text{\hspace{0.17em}⁣}:T\to F$ be a morphism. By [F2] there is an fppf covering $\{{\phi }_j\text{\hspace{0.17em}⁣}:T_j\to T{\}}_{j\in J}$ and morphisms $a_j\text{\hspace{0.17em}⁣}:T_j\to U$ whose images in $F(T_j)$ equal $a{\mathrm{\mid }}_{T_j}$. For each ordered pair the two pullbacks $a_j\circ {\mathrm{p}\mathrm{r}}_1$ and $a_{j^{\mathrm{\prime }}}\circ {\mathrm{p}\mathrm{r}}_2$ on $T_j{\times }_T{T}_{j^{\mathrm{\prime }}}$ agree in $F$, so there are initially fppf-local transition morphisms into $R$. They are unique, since $j=(t,s)$ is a monomorphism, and therefore agree on overlaps and descend to global transition morphisms $r_{j{j}^{\mathrm{\prime }}}\text{\hspace{0.17em}⁣}:T_j{\times }_T{T}_{j^{\mathrm{\prime }}}\to R$ by the represented-sheaf assertion of Fppf sheaves of sets and sheafification, with $t\circ r_{j{j}^{\mathrm{\prime }}}=a_j{\mathrm{p}\mathrm{r}}_1$ and $s\circ r_{j{j}^{\mathrm{\prime }}}=a_{j^{\mathrm{\prime }}}{\mathrm{p}\mathrm{r}}_2$. Put $W_j=a_j^{-1}(W)\subseteq T_j$, open. Then $W_j{\times }_T{T}_{j^{\mathrm{\prime }}}=r_{j{j}^{\mathrm{\prime }}}^{-1}(t^{-1}(W))=r_{j{j}^{\mathrm{\prime }}}^{-1}(s^{-1}(W))=T_j{\times }_T{W}_{j^{\mathrm{\prime }}}$, using the saturation $t(s^{-1}(W))=W$ of step 1.1. Define $W_T={\bigcup }_j{\phi }_j(W_j)$; each ${\phi }_j$ is open by [F3], so $W_T$ is open, and ${\phi }_j^{-1}(W_T)=W_j$: the inclusion $\supseteq$ is clear, while a point $t\in {\phi }_j^{-1}(W_T)$ maps into some ${\phi }_{j^{\mathrm{\prime }}}(W_{j^{\mathrm{\prime }}})$, so the pair $(t,t^{\mathrm{\prime }})\in T_j{\times }_T{T}_{j^{\mathrm{\prime }}}$ lies in $T_j{\times }_T{W}_{j^{\mathrm{\prime }}}=W_j{\times }_T{T}_{j^{\mathrm{\prime }}}$ and hence $t\in W_j$.

$W_T$ represents $T{\times }_F{F}^{\mathrm{\prime }}$. First, the composite $W_T\to T\stackrel{a}{\to }F$ lies in $F^{\mathrm{\prime }}(W_T)$: since $\{W_j\to W_T\}$ is an fppf covering and $F^{\mathrm{\prime }}$ is a sheaf, it suffices to show that each $W_j\to U\to F$ lies in $F^{\mathrm{\prime }}(W_j)$; the morphism $W_j^{\mathrm{\prime }}=W_j{\times }_W{s}^{-1}(W^1){\times }_{W^1}{U}^{\mathrm{\prime }}\to W_j$ is a base change of $t$ and of $g$, hence surjective flat and locally of finite presentation by [F1] and [F3], and the restriction of $W_j\to U\to F$ to $W_j^{\mathrm{\prime }}$ factors through $F^{\mathrm{\prime }}$ by construction, so the sheaf property of $F^{\mathrm{\prime }}$ gives the claim. Conversely, let $f\text{\hspace{0.17em}⁣}:T^{\mathrm{\prime }}\to T$ satisfy $a{\mathrm{\mid }}_{T^{\mathrm{\prime }}}\in F^{\mathrm{\prime }}(T^{\mathrm{\prime }})$. After the fppf base change $\{T^{\mathrm{\prime }}{\times }_T{T}_j\to T^{\mathrm{\prime }}\}$ we may assume $f={\phi }_j\circ f_j$ for some $j$; the condition means that there is an fppf covering $\{{\psi }_i\text{\hspace{0.17em}⁣}:T_i^{\mathrm{\prime }}\to T^{\mathrm{\prime }}\}$ and morphisms $b_i\text{\hspace{0.17em}⁣}:T_i^{\mathrm{\prime }}\to U^{\mathrm{\prime }}$ whose images in the quotient equal those of $a_j{f}_j{\psi }_i$. Refining once more by [F2] supplies morphisms $r_i^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:T_i^{\mathrm{\prime }}\to R$ with $t\circ r_i^{\mathrm{\prime }}=a_j{f}_j{\psi }_i$ and $s\circ r_i^{\mathrm{\prime }}=g{b}_i$, so the image of $f_j{\psi }_i$ lies in $W_j$; hence $f$ factors through $W_T$ fppf-locally, and therefore globally. This proves $T{\times }_F{F}^{\mathrm{\prime }}\cong W_T$.

Conclusion. By step 3.1 every base change of $F^{\mathrm{\prime }}\to F$ along a morphism from a scheme is an open subscheme of the source, so $F^{\mathrm{\prime }}\to F$ is representable by schemes and an open immersion; its image is determined by the saturated open $W$ of step 1.1. If $W=U$, then step 1.2 shows the image of $F^{\mathrm{\prime }}$ in $F$ contains every section of $U$, hence all of $F$ by the sheaf property, and injectivity makes $F^{\mathrm{\prime }}\to F$ an isomorphism. If $g$ is surjective then $W^1=U$ (the image of a surjective morphism is all of $U$) and $W=t(s^{-1}(U))=t(R)=U$ because $t$ is surjective. The Axiom of Choice is used exactly through the quotient-sheaf data of [F2], which enters in steps 1.2-2.1.