I'm trying to get used to the language of generators and relations for C*-algebras through Loring's "Lifting Solutions to Perturbing Problems in C*-Algebras". So far this is what I got from the first few pages.
Let $\mathbf{C}^*$ be the category of C*-algebras, and let $U:\mathbf{C}^*\to\mathbf{Set}$ be the forgetful functor. Given a set of generators $G$ and relations $R$, a representation of $(G,R)$ on a C*-algebra $B$ is an inclusion $b:G\to B$ such that the elements in the image of $b$ satisfy the same relations $R$ as the generators in $G$.
For a fixed set of generators $G$ and relations $R$, define the comma-like category $(G\downarrow U)_R$ to be the category with objects given by pairs $(B,b)$, comprising of a C*-algebra $B$ and a representation $b$ of $(G,R)$ into $B$, and arrows given by $*$-homomorphisms $\pi:B\to B'$ between C*-algebras $B,B'$ such that $b' = \pi\circ b$ is a commuting cone, where $(B,b)$ and $(B',b')$ are representations of $(G,R)$ in the sense above.
Definition (Bounded relation) A set of relations $R$ among a set of generators $G$ is bounded if the category $(G\downarrow U)_R$ makes sense and admits an initial object.
If $R$ is a bounded set of relations among the generators in $G$, then the universal C*-algebra given by the unique (up to isomorphism) initial object in $(G\downarrow U)_R$ is denoted by $C^*\langle G|R\rangle$.
It is well known, and quite easy to see, that $C_0((0,1])$ is the universal C*-algebra for $G=\{h\}$ and $R=\{1\leq h\leq 1\}$, and as stated in the aforementioned book from Loring (Lemma 1.1.1), one can take the function $h(t) = t$ over $[0,1]$ as generator.
My question is then about the proof given there of this fact, which goes like this: "For the relation $0\leq h\leq 1$, every $t\in[0,1]$ gives a representation."
I presume that the word "representation" refers to the notion given above. In this case, what kind of representation do you exhibit for each $t$? Shouldn't you prove that, given an injective representation $\iota$ into the universal C*-algebra, then for every choice of a representation $b$ into another C*-algebra $B$, there exists a unique (up to isomorphism) extension of $b$ to a $*$-homomorphism from the universal C*-algebra to $B$?
