From pg. 56 of Categories for the Working Mathematician:
Definition. If $S:D\rightarrow C$ is a functor and $c$ and object of $C$, a universal arrow ... $u: c \rightarrow Sr$ is universal from $c$ to $S$ when the pair $\langle r, u \rangle$ is an initial object in the comma category $(c \downarrow S)$, whose objects are the arrows $c \rightarrow Sd$. As with any initial object, it follows that $\langle r, u \rangle$ is unique up to isomorphism in $(c \downarrow S)$, in particular, the object $r$ of $D$ is unique up to isomorphism in $D$.
Question: In the textbook, this bolded part is asserted without argument. Why is it true?
Assume $r'$ is another 'such' object in $D$.
What could it mean in this context? That there is an arrow $u':c\to Sr'$ such that $\langle r',u'\rangle$ is an initial object of the comma category.
Since $\langle r,u\rangle$ is also initial, they are isomorphic in the comma category, meaning that there is an isomorphism $d\in D$ making the commutative triangle $Sd\circ u = u'$.