Let $X$ be a scheme and $I \subseteq \mathcal{O}_X$ be a quasi-coherent ideal of finite type. The blowing up $\mathrm{Bl}_I(X)$ has the following universal property: It comes with a morphism $p : \mathrm{Bl}_I(X) \to X$ such that $\langle p^*(I) \rangle \subseteq \mathcal{O}_{\mathrm{Bl}_I(X)}$ is invertible, and for every morphism $f : Y \to X$ such that $\langle f^*(I) \rangle \subseteq \mathcal{O}_Y$ is invertible, there is a unique morphism $\tilde{f} : Y \to \mathrm{Bl}_I(X)$ such that $f = p \tilde{f}$. In other words, $\mathrm{Bl}_I(X) \to X$ is a final object in the category of $X$-schemes which pull back $I$ to an invertible ideal.

First I thought that this implies that $\mathrm{Bl}_I(X)$ is a representing object of the functor

$\mathrm{Sch}^{\mathrm{op}} \to \mathrm{Set},~ Y \mapsto \{f \in \hom(Y,X) : \langle f^*(I) \rangle \subseteq \mathcal{O}_Y \text{ invertible}\},$

but this is wrong: This is not even a functor! Invertible ideals don't pull back to invertible ideals. If $H(Y)$ denotes the set above, then there is a map $\hom(Y,\tilde{X}) \to H(Y)$, which is injective, but far from being surjective.

**Question.** Which functor $\mathrm{Sch}^{\mathrm{op}} \to \mathrm{Set}$ does the blowing up represent? In other words, how can we simplify $\hom(Y,\mathrm{Bl}_I(X))$?

Since we have $\mathrm{Bl}_I(X) = \mathrm{Proj}_X \oplus_{n \geq 0} I^n$ , one can ask more generally what functor $\mathrm{Proj}_X \mathcal{A}$ represents when $\mathcal{A}$ is a sufficiently nice graded $\mathcal{O}_X$-algebra. This is indicated in EGA II, 3.7, but the condition that the partial morphism $r_{\mathcal{L},\psi}$ is defined everywhere is only made explicit for affine $X$ in loc. cit. Cor. 3.7.4. If $\mathcal{A}=\mathrm{Sym}(\mathcal{E})$ for some vector bundle $\mathcal{E}$, then the condition is just that $\psi$ (defined in loc. cit. 3.7.1) is surjective, so that the universal property of $\mathbb{P}(\mathcal{E})$ is very similar to the one of the usual projective space. But $\oplus_{n \geq 0} I^n$ obviously does not have this form, unless $I$ is flat or something like that.

In any case, there seems to be an injective map from

$$\{(f,\mathcal{L},\psi) : f \in \hom(Y,X) , \mathcal{L} \text{ line bundle on } Y, \psi : f^*(\mathcal{A}) \twoheadrightarrow \oplus_n \mathcal{L}^{\otimes n}\}$$

to $\hom(Y,\mathrm{Proj} \mathcal{A})$. EDIT: In Jason Starr's answer it is claimed that this is a bijection. Can someone give a reference in the literature for this observation?

In the note "Elementary Introduction To Representable Functors and Hilbert Schemes" by S. A. Stromme I have found the following amusing exercise:

"Let $X \to S$ be the blowing up of $S$ in some center $Y \subseteq S$. Try to the understand the point functor $h_{X/S}$. Then explain why it is hard to understand blow-ups."

I hope that this does not mean that we cannot understand the functor at all ...

EDIT. The universal property of Proj of a graded quasi-coherent algebra appears in section 16 of "Constructions of schemes" in the Stacks Project (here). It is the usual one when the graded algebra is generated in degree 1. I still wonder if there is any reference in the literature because I want to cite this.