Universal Property of the Universal Line In "An Invitation to Quantum Cohomology" by Kock and Vainsencher, they talk about "the universal line", which is defined as the variety
$U=\{ (L,p)\in Gr(1,\mathbb{P}^r)\times \mathbb{P}^r | p\in L \}$.
This comes with the natural projection onto $Gr(1,\mathbb{P}^r)$ and is a bundle of projective lines over the Grassmanian.
Now my question is, how is the name univesal line justified. Does it have any universal property? E.g. being the universal family for some class of projective space bundles?
If you can help me clarify this, I would be thankful. Also feel free to correct any abuse of terminology in this question. 
 A: The projection map $p:U\to G(1,\mathbb P^r)$ is the universal family of lines, which means that for every family of lines $f:X\to T$ (this means $X$ is closed in $\mathbb P^r\times T$, $f$ is the restriction of the projection and $f^{-1}(t)\subset \mathbb P^r\times\{t\}=\mathbb P^r$ is a line for all $t\in T$) there exists exactly one morphism $\rho_f:T\to G(1,\mathbb P^r)$ such that $X=\rho_f^\ast U$. The name is justified by this universal property, which, put in another way, tells you that $(G(1,\mathbb P^r),p)$ represents the functor $$T\mapsto \{\textrm{ families of lines in }\mathbb P^r\textrm{ parametrized by }T\}.$$ 
If you prefer to phrase it another way: $p$ is the final object in the category of families of lines, where such families are described above, and morphisms between them are cartesian squares. Another name used is tautological family: this reflects the fact that if you look at the fiber of $p$ over a line $[\ell]\in G(1,\mathbb P^r)$ you get exactly $$p^{-1}([\ell])=\ell\subset \mathbb P^r\times \{[\ell]\}=\mathbb P^r.$$
