Thare are questions when I read Daniel Huybrechts' book: The geometry of moduli space of sheaves, page 37, Example 2.2.12. The content I have just copied.
Example 2.1.12 — Let $F_1$ and $F_2$ be coherent $\mathcal{O}_X$-modules on a projective $k$-scheme $X$ and let $E = \text{Ext}_X^1 (F_2, F_1)$. Since elements $\xi \in E$ correspond to extensions
$$0 \to F_1 \to F_\xi \to F_2 \to 0,$$
the space $S = \mathbb{P}(E^\vee)$ parametrizes all non-split extensions of $F_2$ by $F_1$ up to scalars. Moreover, there exists a universal extension
$$0 \to q^* F_1 \otimes p^* \mathcal{O}_S(1) \to \mathcal{F} \to q^* F_2 \to 0$$
on the product $S \times X$ (with projections $p$ and $q$ to $S$ and $X$, respectively), such that for each rational point $[\xi] \in S$, the fibre $\mathcal{F}_\xi$ is isomorphic to $F_\xi$. Indeed, the identity $\text{id}_E$ gives a canonical extension class in $E^\vee \otimes_k E = \text{Ext}_X^1 (F_2, E^\vee \otimes_k F_1)$. Let $\pi$ denote the canonical homomorphism $E^\vee \otimes \mathcal{O}_S \to \mathcal{O}_S(1)$ and consider the class $\pi_*(\text{id}_E)$, i.e. the extension defined by the push-out diagram
$$\begin{matrix} 0 & \longrightarrow & E^\vee \otimes q^* F_1 & \longrightarrow & \mathcal{q^*G} & \longrightarrow & q^* F_2 & \longrightarrow & 0 \\ & & \downarrow{\scriptstyle\pi \otimes 1} & & \downarrow & & \Vert & & \\ 0 & \longrightarrow & p^* \mathcal{O}_S(1) \otimes q^* F_1 & \longrightarrow & \mathcal{F} & \longrightarrow & q^* F_2 & \longrightarrow & 0 \end{matrix}$$
where the extension in the top row is given by $\text{id}_E$. Note that $\mathcal{F}$ is $S$-flat for the obvious reason that $q^* F_1$ and $q^* F_2$ are $S$-flat.
$\cdot$ The first question: Why the pullback of his construction "universal extension" along $[\xi]$ is the extension that corresponding to $\xi$?
His construction of $\mathcal{U}$, the universal extension $\pi_*(\mathrm{id}_E)$, is just the composition:
\begin{align*}
E^\vee \otimes E &= \text{Ext}_X^1(F_2, E^\vee \otimes F_1) \rightarrow \text{Ext}_{S \times X}^1(q^*F_2,E^\vee\otimes q^*\mathcal{F}_1) \rightarrow \text{Ext}_{S \times X}^1(q^*F_2,p^*\mathcal{O}_S(1)\otimes q^*F_1)
\end{align*}
Here the first arrow is applying $q^*$, which is flat any way, the second arrow is just the $\pi_*$ he has mentioned. Now we pullback along $[\xi]$, it is something like pulling back items from the last term to the first term, but how? Why the result is exactly $\xi$?
$\cdot$ The second question: What is his purpose of choosing the tautological bundle? Is there the more global way to describe the functor $h_S$ such that we can extended the point $\xi$ to any $T\rightarrow S$, $\mathrm{i.e.}$ view $\mathrm{Ext}$ as a functor?
Thank you for your answer!!!!
