Let $S$ be a scheme, $\mathcal{E}$ a locally free $\mathcal{O}_S$-module of finite rank $r$ and $\pi:P=Proj(\mathcal{E})\to S$ the projective bundle. There is a canonical surjective map of sheaves $\psi:\pi^*\mathcal{E}\to \mathcal{O}(1)$.

Now for all $S$-schemes $f:X\to S$, there is a bijection (functorial in $X$) between the set of maps of $S$-schemes $X\to P$ and the set of equivalence classes of pairs $(\mathcal{L},\varphi)$ where $\mathcal{L}$ is an invertible sheaf on $X$ and $\varphi:f^*\mathcal{E}\to \mathcal{L}$ is a surjective map of sheaves. The equivalence relation is that $(\mathcal{L}',\varphi')\sim (\mathcal{L},\varphi)$ if and only if there exists an isomorphism $\tau:\mathcal{L}\to\mathcal{L}'$ such that $\varphi'=\tau\circ\varphi$.

For $X=P$ and $f=\mathrm{id}_P$, one gets a universal equivalence class which is supposed to be that of $(\mathcal{O}(1),\psi)$. So is $(\mathcal{O}(1),\psi)$, or only its equivalence class, canonical ?

uniqueisomorphism, because your $\tau$ isunique: an automorphism of $\mathcal{L}$ dominated by the identity on $f^{\ast}E$ must be the identity. Put another way, the pairs being classified here have no nontrivial automorphisms, so this is a moduli problem for which nothing confusing about isomorphisms messes it up in the end. – BCnrd Sep 25 '10 at 8:06