I have been trying to get used to working with representable functors in place of getting my hands dirty with schemes. In particular I am working on a problem involving blowing up over an arbitrary noetherian scheme X. Let $\mathscr{I} \subseteq \mathcal{O}_{X}$ be a coherent sheaf of ideals. Let $\widetilde{X}$ be the blowing up along $\mathscr{I}$ and let $\mathbb{P}(\mathscr{I})$ be the projective bundle associted to $\mathscr{I}$. I want to think of these simply as objects representing sheaves on the big Zariski site $\text{Sch}/X$.
If we choose an affine cover $\{ U_{i} = \operatorname{Spec}A_{i} \}$ of $X$ then over each $A_{i}$ these schemes look like $\operatorname{Proj}(A_{i} \oplus \mathscr{I}(U_{i}) \oplus \mathscr{I}(U_{i})^{2} \oplus \cdots )$ and $\operatorname{Proj}(\operatorname{Sym}_{A_{i}}(\mathscr{I}(U_{i})))$.
I want to understand how $X$-morphisms $\widetilde{X} \rightarrow \mathbb{P}(\mathscr{I})$ arise by gluing universal properties for the projective spaces over each affine.
My understanding is that the key theorem towards this is the gluing theorem in the stacks project here. My main question here is to clarify that I am understanding this result correctly. My understanding is as follows:
Define sheaves on the big Zariski site $\text{Sch}$ (not over $X$) as follows, $$ \hom(-, \widetilde{X}): \text{Sch} \longrightarrow \text{Set} , \\ \hom(-, \mathbb{P}(\mathscr{I})): \text{Sch} \longrightarrow \text{Set} $$ Then we pullback these sheaves to the localized sites $\text{Sch}/U_{i}$ and assume we have morphisms, $$ \phi_{i}: \hom(-, \widetilde{X})|_{\text{Sch}/U_{i}} \longrightarrow \hom(-, \mathbb{P}(\mathscr{I})|_{\text{Sch}/U_{i}} $$ which agree on fibered products (read: intersections). Then the lemma gives us a morphism of sheaves on $\text{Sch}/X$, $$ \hom(-, \widetilde{X})|_{\text{Sch}/X} \longrightarrow \hom(-, \mathbb{P}(\mathscr{I})|_{\text{Sch}/X} $$ First of all, is this actually the correct interpretation of the lemma? It just seems odd to me that in order to obtain a morphism over $X$ we have to begin with sheaves on the category of schemes not over $X$. And assuming my interpretation is correct, it is still not completely clear to me that $$ \hom(-, \widetilde{X})|_{\text{Sch}/U_{i}} = \hom_{U_{i}}(-, \widetilde{X} \times_{X} U_{i}) $$ Is that actually true? Obviously I would like that to be true in order to appeal to the mapping properties for Proj over an affine base.