Let $X$ be an algebraic space and $L,M$ are vector bundles with rank $n,m$. Then, It is known that $\mathbb{P}Ext^1(L,M)$ is a parameter space for isomorphism classes of vector bundle of rank $n+m$, which is obtained by extension of $L$ by $M$.

My Question is, What is moduli functor $\mathbb{P}Ext^1(L,M)$ represents?.

In the book of Huybrechts & Lehn, there is a familly, which they call "universal familly" $\mathcal{U}$ on X $\times$ $\mathbb{P}Ext^1(L,M)$, that is

$0 \to (\pi_1)^*M\otimes(\pi_2)^*\mathcal{O}(1) \to \mathcal{U} \to (\pi_1)^*L \to 0$.

So, my second question is that is this really a universal family for moduli functor that $\mathbb{P}Ext^1(L,M)$?.

Thank you for read my questions.