Let $F: (Sch/S)^{op} \to Sets$ be a functor that is both a sheaf in the Zariski topology and has an open covering $(f_i: F_i \to F)_{i \in I}$, where each of the $F_i$ is representable.
In theorem 8.9 of Görtz-Wedhorn's book on algebraic geometry, it is proven that the functors $F_i$ glue to a scheme $X$. Then, because $F$ is a sheaf, the morphisms $f_i: F_i \to F$ glue to a morphism $f: X \to F$ – which the authors claim to be an isomorphism. Is there an elementary way to see that $f$ is an isomorphism? The book doesn't use universal objects, so an answer that doesn't address them would be great.
Best Answer
From the construction of $X$, we have "inclusions" $F_i \to X$. These glue to give a morphism $F \to X$, which is the inverse of the morphism $X \to F$ that you described.