Let's suppose we have two vector spaces $V$ and $W$. Then, the tensor product $V\otimes W$ is determined by the following universal property: Given any vector space $Z$ and bilinear map $h\colon V\times W\to Z$, there is a unique linear map $\widetilde{h}\colon V\oplus W\to Z$ yielding the following commutative diagram. (Here $\varphi$ is the map given by $\varphi(v,w)=v\otimes w$.)
My question concerns universal C$^{*}$-algebras. Consider for instance the universal C$^{*}$-algebra generated by a single unitary $v$; this is well known to be $C(\mathbb{T})$. In each book that I have checked, the universal property is stated as follows: If $A$ is a C$^{*}$-algebra containing a unitary $u$, then there is a unique $*$-homomorphism $\psi\colon C(\mathbb{T})\to A$ such that $ \psi(v)=u$.
I would like to know how to translated the above statement (or understand how it comes from) a universal property with a commutative diagram as, for instance, the tensor product did above.
There is a function from $\{v\}$ into the unitary elements of $C(\mathbb T)$ such that for any such function into the unitaries of another algebra $C$ there exists a unique $C^*$ map filling those two function to a triangle.
The story this should fit into more generally is that there's a left adjoint to a forgetful functor that sends a $C^*$ algebra to its set (maybe group?) of unitaries.