I want to prove that the free product $A*B$ of two unital $C^*$-algebras $A$ and $B$ is a coproduct in the sense of category theory. Remember the construction of $A*B$: Take generators $\{a:a\in A\}\cup\{b:b\in B\}$ and let $A*B$ the $C^*$-algebras generated by this set. Then the inclusion maps $\iota_A$ and $\iota_B$ are obvious and are $*$-homomorphisms. Now let $X$ be another $C^*$-algebra with *-homomorphisms $\mu:A\rightarrow C$ and $\nu:B\rightarrow C$, then we can define $u:A*B\rightarrow C$ by $u(a)=\mu(a)$ and $u(b)=\nu(b)$, then this extends to a $*$-homomorphism $u:A*B\rightarrow C$. My question is: why is this map the unique map such that $u\circ\iota_A=\mu$ and $u\circ\iota_B=\nu$? Can someone help me?
Thank you very much.
Since $u\circ i_A=\mu$, we have $u(i_A(a))=\mu(a)$ for all $a\in A$; hence $u(a)=a$ (if, as you do, you identify $A$ with its image $i(A)$). At the same time, $u(b)=\nu(b)$ for $b\in B$. Thus, $u$ is uniquely determined on $A\cup B\subset A*B$, which uniquely determines $u$ on the entire $A*B$.