The Stacks project

Is the second condition actually the sheaf $h_S\times F'$ is an algebraic space over $S$? Since the via the identification $Sh((Sch/S) _ {fppf}) \cong Sh((Sch/S') _ {fppf})/h_S$ the functor $j^{-1}:Sh((Sch/S') _ {fppf})\rightarrow Sh((Sch/S) _ {fppf})$ is identified with $Sh((Sch/S') _ {fppf}) \ni F\mapsto (h_S\times F\rightarrow h_S) \in Sh((Sch/S') _ {fppf})/h_S$ according to Lemma 7.25.9. Maybe the citation of Lemma 7.25.4 should also be Lemma 7.25.9?

I think the lemma is OK as it stands. The point of the lemma is that in checking that the restriction of $F'$ to $(Sch/S)$ is an algebraic space, it suffices to prove the functor $h_S \times F'$ on the category of schemes over $S'$ is an algebraic space. Note that it scarcely makes sense to consider the functor $h_S$ on the category of schemes over $S$.