Every representable functor is an algebraic space

The sheaf condition. $h_T$ is an fppf sheaf by [F1]: a morphism $T^{\mathrm{\prime }}\to T$ is determined by its restrictions to an fppf covering of $T^{\mathrm{\prime }}$ and such restrictions glue uniquely, which is exactly the sheaf condition for the represented functor.

The diagonal. The diagonal $h_T\to h_T\times h_T$ corresponds under the Yoneda identification $h_T\times h_T\cong h_{T{\times }_{S}T}$ of [F3] to the morphism $h_T\to h_{T{\times }_{S}T}$ induced by the diagonal $T\to T{\times }_{S}T$. To test representability, let $\xi \text{\hspace{0.17em}⁣}:Z\to h_T\times h_T$ be a morphism from a scheme $Z$, corresponding to a morphism $Z\to T{\times }_{S}T$; the fibre product $h_T{\times }_{h_T\times h_T}Z$ is then represented by the fibre product $Z{\times }_{T{\times }_{S}T}T$, which is a scheme by [F3]. Hence the diagonal is representable by schemes.

The etale cover. Condition 3 of [F2] is satisfied by the identity $h_T\to h_T$: it is representable, because for any $\xi \text{\hspace{0.17em}⁣}:Z\to h_T$ the fibre product $h_T{\times }_{h_T}Z\cong Z$ is a scheme, and it is etale and surjective because the identity of $T$ is etale and surjective by [F4] and representability is witnessed by the identity base changes. Hence $h_T$ is an algebraic space.

Full faithfulness. For $S$-schemes $T,T^{\mathrm{\prime }}$ and a natural transformation $\alpha :h_T\to h_{T^{\mathrm{\prime }}}$ of the presheaves in [F4], put $g={\alpha }_T({\mathrm{id}}_T)\in {\mathrm{Mor}}_S(T,T^{\mathrm{\prime }})$. For every $S$-scheme $Z$ and $f:Z\to T$, naturality along $f$ gives ${\alpha }_Z(f)=h_{T^{\mathrm{\prime }}}(f)({\alpha }_T({\mathrm{id}}_T))=g\circ f$, since $h_T(f)({\mathrm{id}}_T)=f$. Thus $g$ determines every component of $\alpha$. Conversely any $S$-morphism $g:T\to T^{\mathrm{\prime }}$ defines the natural transformation $f\mapsto g\circ f$, because precomposition commutes with this formula; evaluating at ${\mathrm{id}}_T$ recovers $g$. These constructions are inverse, proving the required hom-set bijection without importing a full-faithfulness theorem from the representation definition. Together with step 1.3 this embeds schemes fully faithfully into algebraic spaces.