Let $A$ be a $C^*$-algebra and let $(A_n)$ be an increasing sequence of $C^*$-subalgebras of $A$ whole union is dense in $A$.Let $\phi_n:A_n\to A_{n+1}$ be the inclusion map.Then $A$ is $*$ isomorphic to the direct limit of the direct sequence $(A_n,\phi_n)$.
Suppose $B=lim(A_n,\phi_n)$,how to show that $A$ is $*$ isomorphic to $B$?
Maps out of both $A$ and $B$ are uniquely determined by their restrictions to $\cup_n A_n$, via density. So there are unique maps $A\to B$ and $B\to A$ restricting to the identity on $\cup_n A_n$; the composites give morphisms $A\to A$ and $B\to B$ which fix the dense subspaces $\cup_n A_n$, which are thus the respective identities.