# Representability criterion for schemes

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.