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I am trying to understand. Lemma 25.15.4

Let $F$ be a contravariant functor the category of schemes with values in the category of sets. Suppose that

  • $F$ satisfies the sheaf property.
  • There exists set $I$ and subfunctors $F_i \subseteq F$

such that

  • each $F_i$ is representable.
  • each $F_i$ is representable by open immersions.
  • the collection $F_i$ covers $F$.

Then $F$ is representable.

The definitions of subfunctors, representable by open immersions are cover are quite long and I refer to the link.

There is one particular step in the given proof that I do not follow.

We have to show that $\varphi_{ij}^{-1}(U_{ji} \cap U_{jk}) = U_{ij} \cap U_{ik}$. This is true because (a) $U_{ji} \cap U_{jk}$ is the largest open $U_{ji}$ such that $\xi_j$ restricts to an element of $F_k$, (b) $U_{ij} \cap U_{ik}$ is the largest open of $U_{ij}$ such that $\xi_i$ restricts to an element of $F_k$. (c) $\varphi^*_{ij} \xi_j = \xi_i$.

Firstly, I do not know how (a) and symmetrically, (b) holds. Even with (c), I do not understand how the deduction follows.

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