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.