Can the choice of definition of morphisms for a slice category be justified categorically? An example of a slice category $(\mathscr{C} \downarrow c)$ derived from some fixed object $c$ of some base category $\mathscr{C}$, would be one whose objects correspond to the $\mathscr{C}$-morphisms $f$ having $\mathrm{cod}(f) = c$, and whose morphisms $f \to g$ correspond to the $\mathscr{C}$-morphisms $h:\mathrm{dom}(f) \to \mathrm{dom}(g)$ such that $g \circ h = f$. 1
This question is specifically about this particular construction of the morphisms for a slice category.
This construction is certainly reasonable enough, but in my experience Category Theory is all about giving such reasonable-looking constructions a rigorous justification in terms of some universality property or other.
Can such a justification be given for the standard definition of morphisms in a slice category?

1 As usual, a dual definition can be obtained by reversing the arrows.
 A: For any categories $\mathscr{C}$ and $\mathscr{D}$, there is the functor category $\mathscr{C}^\mathscr{D}$ whose objects are functors $F:\mathscr{D}\to\mathscr{C}$ and whose morphisms are natural transformations $\alpha:F\Rightarrow G$ (though one has to be set-theoretically careful if $\mathscr{D}$ is not small).
Let $\mathcal{I}$ be the category $\{0\xrightarrow{t}1\}$, i.e. there are two objects $0$ and $1$, and one non-identity morphism $t:0\to 1$. Clearly, given a category $\mathscr{C}$, functors $F:\mathcal{I}\to\mathscr{C}$ correspond to $\mathscr{C}$-morphisms, with $0$ going to the domain and $1$ going to the codomain. A natural transformation $\alpha:F\Rightarrow G$ between two such functors $F$ and $G$ is just a choice of $\mathscr{C}$-morphisms $\alpha_0$ and $\alpha_1$ making this diagram commute:
$$\require{AMScd}
\begin{CD}
F(0) @>\alpha_0>> G(0)\\
@VF(t)VV  @VVG(t)V\\
F(1) @>>\alpha_1> G(1)
\end{CD}$$
Thus the functor category $\mathscr{C}^{\mathcal{I}}$ is (isomorphic to) the category whose objects are all $\mathscr{C}$-morphisms $f:a\to b$, and for which a morphism $\alpha$ from $f:a\to b$ to $g:c\to d$ is a pair of morphisms $\alpha_0:a\to c$ and $\alpha_1:b\to d$ such that
$$\require{AMScd}
\begin{CD}
a @>\alpha_0>> c\\
@VfVV  @VVgV\\
b @>>\alpha_1> d
\end{CD}$$
There is a natural functor $\mathrm{cod}:\mathscr{C}^\mathcal{I}\to\mathscr{C}$ that sends a $\mathscr{C}^\mathcal{I}$-object $f:a\to b$ to its codomain, the $\mathscr{C}$-object $\mathrm{cod}(f)=b$, and sends a $\mathscr{C}^\mathcal{I}$-morphism $(\alpha_0,\alpha_1):f\to g$ to the $\mathscr{C}$-morphism $\alpha_1:\mathrm{cod}(f)\to \mathrm{cod}(g)$.
Then the slice category $(\mathscr{C}\downarrow c)$ is just the subcategory of $\mathscr{C}^\mathcal{I}$ whose objects $f$ satisfy $\mathrm{cod}(f)=c$ and whose morphisms satisfy $\mathrm{cod}(a_0,a_1)=\mathrm{id}_c$. This should just be equivalent to taking the pullback of
$$\require{AMScd}
\begin{CD}
 & & \mathscr{C}^{\mathcal{I}}\\
& &  @VV\mathrm{cod}V\\
\{c\circlearrowleft\mathrm{id}_c\} @>>\mathrm{inclusion}> \mathscr{C}
\end{CD}$$
in the category of categories.
So ultimately, this is a long way of saying that the choice of morphisms in the slice category boils down to the way that a natural transformation is defined, since a slice category can be seen as a particular subcategory of $\mathscr{C}^\mathcal{I}$.
