# Details of the criterion for representability of a functor of S-schemes

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 $$\text{Sch}/S$$.