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I've been trying to prove a fairly classical, well-known result, but am running into a lot of trouble following any of the proofs I have found. At the moment I am following EGA I 0.4.5.4.

Let $F: \text{Sch}^{\text{op}} \rightarrow \text{Set}$ be a contravariant functor on the category of schemes with values in sets. Suppose we have a family of subfunctors $\mu_{i} : F_{i} \hookrightarrow F$ of $F$ satisfying the following:

i) The subfunctors $F_{i}$ are representable by open immersions;

ii) The functor $F$ satisfies the sheaf condition on the Zariski topology;

iii) Each $F_{i}$ is representable

iv) The open subfunctors $F_{i}$ cover $F$.

I want to show that $F$ is representable by a scheme.

The basic idea seems to be not so difficult. We first obtain gluing data, and then show that the resulting glued scheme represents $F$. Say $F_{i}$ is represented by $(X_{i}, \eta_{i})$ where $\eta_{i} \in F_{i}(X_{i})$ is the universal family. We can form the fibered product \begin{equation} \label{square} F_{ij} = F_{i} \times_{F} F_{j} \end{equation} which is itself representable since $F_{i}$ and $F_{j}$ are representable by open immersions. Moreover, if $u_{ji}$ is the pullback of $\mu_{j}$ and $u_{ij}$ is the pullback of $\mu_{i}$, then the corresponding morphisms via Yoneda's lemma, \begin{align} \phi_{ij}: X_{ij} \longrightarrow X_{j} \\ \phi_{ji}: X_{ij} \longrightarrow X_{i} \end{align} are open immersions. Suppose $F_{ij}$ is represented by $(X_{ij}, \eta_{ij})$.

So far I think I understand. But then Grothendieck supposes we have a morphism $\nu: Y \rightarrow X_{ij}$. He uses this to deduce the following, \begin{align} (u_{ji}(Y))(F(\nu) (\nu_{ij})) = F(\phi_{ji} \circ \nu)(\nu_{i}) \\ (u_{ij}(Y))(F(\nu) (\nu_{ij})) = F(\phi_{ij} \circ \nu)(\nu_{j}) \end{align} How does he obtain this? So the morphism $Y \rightarrow X_{ij}$ will provide a morphism $\hom(-, Y) \rightarrow \hom(-, X_{ij})$. This then provides a cone over the pullback square \eqref{square}. But I can't actually see how those identities are equivalent to the universal property, which seems to be what he is claiming.

More generally, is anyone able to point me to a more accessible proof of this. I'm having trouble following the only other one I know of , which is in the Stacks Project.

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