According to Wikipedia article on universal property,
If $U:D \rightarrow C$ is a functor and $X$ an object in $C$, then initial morphism from $X$ to $U$ is an object in category $(X\downarrow U)$ of morphisms from $X$ to $U$. In other words, it consists of a pair $(A,\Phi )$ where $A$ is an object of $D$ and $\Phi:X\rightarrow U(A)$ is a morphism in $C$.
My question is why do we even need to start with category $D$ and functor $U$? Why can't we just have the universal property with just category $C$ and it's Arrow category, since we already have everything we need in category $C$?
Here is an example to make my question more clear:
Suppose we have category $C$ with following morphisms:
$f_1: x \rightarrow U(z)$
$f_2: x \rightarrow U(b)$
$f_3: x \rightarrow U(a)$
$U(g): U(a) \rightarrow U(b)$
$U(h): U(a) \rightarrow U(z)$
Then we will have the following as the Arrow category of $C$:
$U(g): <x,f_3,U(a)> \rightarrow <x,f_2,U(b)>$
$U(h):<x,f_3,U(a)> \rightarrow <x,f_1,U(z)>$
And it can be seen that $<x,f_3,U(a)>$ is an initial object in Arrow category, hence making $f_3$ an initial morphism.