I do not understand one argument in the proof of Proposition 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!

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    $\begingroup$ I am not sure this necessarily answers your question but maybe have a look at the link: stacks.math.columbia.edu/tag/01JJ $\endgroup$ – user631975 Mar 26 '19 at 14:52
  • $\begingroup$ Unfortunately, this does not really help because the definition of representability by an open immersion in your link (if one reformulates it) leads immediately to the injectivity of the described map. The definition in EGA I seems to be much weaker. $\endgroup$ – Daniel W. Mar 26 '19 at 15:49
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    $\begingroup$ Please do not post a question simultaneously on MO and MSE. See this Meta Math post and its MO version for more info. $\endgroup$ – Arnaud D. Mar 26 '19 at 16:36
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    $\begingroup$ Which version of EGA I are you talking about? Can you give a link? $\endgroup$ – user170039 Mar 26 '19 at 18:06
  • $\begingroup$ @user170039 I'm guessing they mean the 1971 version published by Springer (which is not freely available, as far as I know), rather than the original version published in 1960 by the IHÉS. $\endgroup$ – Richard D. James Mar 27 '19 at 2:49

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