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!