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I've come across a problem that's made me look back over representabiilty of scheme functors and I'm having a lot of trouble piecing together some categorical details that I used to think I understood. To give ourselves a concrete example with which to work, let $S$ be a noetherian separated scheme and let $\mathcal{E}$ be a quasicoherent sheaf of $\mathcal{O}_{S}$-algebras.

I would like to show that there is a scheme $\bf{Spec}(\mathcal{E}) \rightarrow S$ via a representability criterion.

Let $F: \text{Sch}/S \rightarrow \text{Set}$ be a contravariant functor defined as follows: On objects it sends $(T \stackrel{\mu}{\rightarrow}S)$ to the set $\text{Hom}_{\mathcal{O}_{S}-alg}(\mathcal{E}, \mu_{*} \mathcal{O}_{T})$. To show it is representable it is enough to show that it is a sheaf for the big Zariski site on $S$, and that it is locally representable.

Let's assume we have shown already that it is a Zariski sheaf. I want to clarify exactly what more must be shown.

We begin with a morphism of functors, $$ \eta: F \longrightarrow \text{Hom}_{S}(-, S). $$ Now firstly, am I correct that the functor $\text{Hom}_{S}(-, S)$ will an object $(T \stackrel{\mu}{\rightarrow} S)$ and simply return that same object?

Now let $\{ U_{i} \}$ be an affine cover of $S$ with inclusions $\tau_{i}: U_{i} \hookrightarrow S$. We want to show that the pulled back functors, $\require{AMScd}$ \begin{CD} F_{i} @>{}>> F\\ @VVV @VVV{\eta}\\ \text{Hom}_{S}(-, U_{i}) @>{\tau_{i}^{*}}>> \text{Hom}_{S}(-, S) \end{CD} are representable. But I'm not even sure what this pullback is supposed to be. In terms of functors, a pullback is supposed to define a functor from slice categories $$ \mathcal{C}/a \longrightarrow \mathcal{C}/b $$ based on a morphism $b \rightarrow a$. But here my morphism is a morphism of functors $$\text{Hom}_{S}(-, U_{i}) \stackrel{\tau_{i}^{*}}{\rightarrow} \text{Hom}_{S}(-, S).$$

So what category precisely is the pulled back functor $F_{i}$ defined on? I would like it to be the category $\text{Sch}/U_{i}$ so I can then apply the known construction to the affine case. But I don't see why from a purely formal viewpoint the functor should be defined on that category rather than just still on $\text{Sch}/S$.

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