I am reading through Rordam's C$^{*}$-algebra book and I think I am misunderstanding something. On page 92, Rordam defines Inductive limits in an arbitrary category as follows.
He goes on to show that the uniqueness of the map $\lambda\colon A\to B$ actually implies that there is a unique isomorphism $\lambda\colon A\to B$ making (6.4) commute. So, applying this to the category of C$^{*}$-algebras, I assumed this to mean that there is a unique isometric $*$-isomorphism $\lambda\colon A\to B$, provided that $(B, \{\lambda_{n}\}_{n=1}^{\infty})$ is a system as given in (ii). However, I think I am really not understanding something here because on page 94, Rordam has the following proposition.
If my understanding was correct (which it obviously isn't) then $\lambda$ as in 6.2.4. would be an isomorphism and, hence, a bijection. But the way the proposition is phrased, it is clear that this is not always the case. I would really appreciate if someone would help me understand what is happening here.
Thank you.
The unique morphism you find has not to be an isomorphism. Trivial examples are easily constructed. Take for example $A_n = \mathbb C$ with the identity as connecting map and $B = M_2(\mathbb C)$, where you map $\mathbb C $ as diagonal into $M_2(\mathbb C)$.
That gives you a map from the limit $\mathbb C$ into $M_2(\mathbb C)$ which clearly cannot be an isomorphism.