I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I.
When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ by gluing the objects $X_i$ representing the subfunctors $F_i$, Grothendieck shows for an arbitrary $S$-ringed space $T$, that there is a bijection of the form $Hom_S(T,X) \to F(T), g \mapsto F(g)(\xi)$.
For the proof of the injectivity, he needs the argument that the fibre product $F_i \times_F h_X$, where $h_X \to F$ is the natural transformation corresponding to $\xi$, is representable by $(Z_i,(\xi_i', \rho_i'))$, where $Z_i$ is isomorphic to $X_i$, $\rho_i':Z_i \to X$ is the canonical injection and $\xi_i' = F(\rho_i')(\xi)$. For this fact, he argues with condition (i) in the Theorem, i.e. that $F_i \to F$ is representable by an open immersion.
I cannot see how the representability of the fibre product by this tuple follows from this.
Thank you for your help!