= Solution
Starting from $(r,e)$, the construction in (iii) preserves the composite to the <affine line> and sends $Z_A$ to the $A$ zero section and $Z_B$ to the $B$ zero section. Part (ii) therefore recovers $(r,e)$.
Conversely, start with $u:T\to X$ and construct $(r,e)$. The only points of $X$ outside $A$ lie on $B_0$, and the only points outside $B$ lie on $A_0$. Consequently $u^{-1}(A)=T\setminus Z_B=T_A$ and $u^{-1}(B)=T\setminus Z_A=T_B$. On each of these <open subsets>, $u$ is the unique <morphism of schemes> to the appropriate <affine line> with coordinate $r$. Thus part (iii) recovers $u$.
Finally, a <ring homomorphism> $R\to R'$ sends $r$ to $r'$ and induces $R/(r)\to R'/(r')$, sending $e$ to an <idempotent> $e'$. Inverse images of the <clopen sets> $Z_A,Z_B$ are precisely those determined by $e'$. The constructions therefore commute with this <base change>. We have a <natural bijection>
$$
\boxed{\operatorname{Hom}_{\mathsf{Sch}}(\operatorname{Spec}R,X)\cong F(R),}
$$
so $F$ is a <functor represented by a scheme>, represented by the <affine line with doubled origin>. The representing <scheme> is not required to be an <affine scheme>.
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