Gluing algebraic spaces along open subfunctors

Disjoint unions. Interpret $G={\coprod }_i{F}_i$ as the coproduct in fppf sheaves. Explicitly $G(T)$ consists of a decomposition $T={\coprod }_i{T}_i$ into disjoint open-and-closed subschemes, together with $x_i\in F_i(T_i)$. This formula is a sheaf: a matching local decomposition descends by taking images of its pieces along the covering maps, which are open; the cocycle makes these images disjoint and makes each piece upstairs the inverse image of the descended piece. The complement is the union of the other open images, hence each descended piece is also closed. The matching sections then glue uniquely in each $F_i$. A morphism from this sheaf to any sheaf is uniquely specified by its restrictions to the $F_i$, because the decomposition is a Zariski cover; thus the formula has the coproduct universal property. Choose $U_i\to F_i$ using AC and put $U={\coprod }_i{U}_i$, a scheme by disjoint affine-chart gluing [F4]. Over $(T_i,x_i)\in G(T)$ the pullback of $U\to G$ is the scheme ${\coprod }_i(T_i{\times }_{F_i}{U}_i)$, etale and surjective over $T$. For two sections of $G(T)$, their equality locus over $T_i\cap T_j^{\mathrm{\prime }}$ is empty when $i\mathrm{\ne }j$ and is the scheme equality locus in $F_i$ when $i=j$, represented by its diagonal. These schemes form a disjoint union over the disjoint open-and-closed pieces of $T$; it represents the diagonal pullback. Hence $G$ has a representable diagonal and the required etale scheme cover, so it is an algebraic space.

The cover for open gluing. Assume (2). Choose etale scheme covers $U_i\to F_i$ and their disjoint union $U$. The composites $U_i\to F$ are representable and etale by [F2], since $F_i\to F$ are representable open immersions. Their union is representable: over $T\to F$, its fibre product is the disjoint union of the schemes $T{\times }_F{U}_i$, formed by [F4]. It is etale componentwise and surjective, because sections of $F$ locally land in some $F_i$ by the given sheaf surjectivity and then locally lift to $U_i$. Thus $U\to F$ is a representable etale surjective cover.

The diagonal for open gluing. Given two sections $x,y\in F(T)$, let $T_i=x^{-1}(F_i)$ and $T_j^{\mathrm{\prime }}=y^{-1}(F_j)$, which are open subschemes by representability of the inclusions. The two families cover $T$: the hypothesis supplies local lifts, and the images of covering morphisms cover the underlying scheme. On $T_i\cap T_j^{\mathrm{\prime }}$, any equality $x=y$ forces $y$ to land in $F_i$. The locus where $y$ lands in $F_i$ is an open subscheme; on it both sections lie in $F_i$, and their equality is represented by the diagonal of $F_i$. This scheme is precisely the equality functor on $T_i\cap T_j^{\mathrm{\prime }}$. These representing schemes agree canonically on base overlaps, their overlap maps are open immersions, and their canonical identifications satisfy the cocycle identity. They glue by [F4] to a scheme representing the equality functor on $T$, since a compatible family of maps glues uniquely. Therefore every scheme base change of ${\mathrm{\Delta }}_F$ is a scheme. Together with step 2.1 and the assumed sheaf condition, this proves that $F$ is algebraic. AC selects covers and is inherited from the suppliers.