## EGA I (Springer), Proposition 0.4.5.4

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!