In a slice category C/A of a category C over a given object A, What is the role of the identity morphism of A in C with respect to C/A In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of  the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition?  
I understand that as an arrow targeting $A$, it is an object of $C/A$.  However, in $C$, $1_A$ is not practically composable, because it is an identity morphism and $(X\to A) \circ 1_A$ is just reduced to $X\to A$.  This doesn't seem like it would necesarily be the case in $C/A$, because in $C/A$, $1_A$ is not the identity morphism (the commutative diagram that represents $1_A\to 1_A$ should play this role).  
It seems possible that $1_A\to 1_A$ could be reduced to $1_A$, but I don't know if this is the case and how I would validate it.  If it isn't the case, then it would seem that for every other object, $Z\to A$ in $C/A$, at least two morphisms would exist: #1, $(Z\to A) \to (Z\to A)$, (the identity morphism in $C/A$); and, #2, $(Z\to A)\to (A\to A)$.
I haven't found such an explanation in the books/articles I've read, which leads me to believe I am misunderstanding something fundamental about how slice categories are derived, but without an authoritative reference, I can't be sure.
 A: Objects of $C/A$ are pairs $(X \in C,y : X \to A)$
Maps in $C/A$ between $f : (X \in C,y : X \to A) \to (Z \in C,w : Z \to A)$ are maps $f : X \to Z$ which make the triangle commute: $y = wf$
So $(A \in C, 1_a : A \to A)$ is an object in $C/A$, and $1_a : (A \in C, 1_a : A \to A) \to (A \in C, 1_a : A \to A)$ is a map.

Some comments stated $(A \in C, 1_a : A \to A)$ is the terminal object, here's the proof:
take an object $(X \in C,y : X \to A)$
and now $y$ can be seen as a map $(X \in C,y : X \to A) \to (A \in C, 1_a : A \to A)$
A: In $C/A$, $1_A$ is a final object in the category - for every object $f:X\to A$ in $C/A$ there is a unique morphism $f\to 1_A$ - that is, $\hom(f,1_A)$ is always a singleton.
You are confusing $1_A$, with $1_{1_A}$. $1_A$ is an object of $C/A$. $1_{1_A}$ is the identity morphism for that object. That morphism is always trivial with composition, as are all identity morphisms.
It might be useful to consider the basic case of $C=\text{Set}$.  $\text{Set}$ has final objects equal to the singleton sets.
Given a set $A=\{1,2\}$, it turns out that $\text{Set}/A$ is pretty much $\text{Set}\times\text{Set}$. The sets corresponding to $f:X\to A\in \text{Set}/A$ are $f^{-1}(1)$ and $f^{-1}(2)$. Now, if $C_1,C_2$ both have final objects, then $C_1\times C_2$ has a final object. So the final object in $\text{Set}^2$ is just two pairs of singleton sets. But it we reverse the operation, we see that corresponds to an element of $f:X\to A$ which is $1-1$ and onto.
For more general sets $A$, there is a sense in which $\text{Set}/A$ is equivalent to something we can write as $\text{Set}^A$. Again, we see that the final objects must be final at every "point" $a\in A$, and thus that the final objects of $\text{Set}^A$ are just lists of singletons indexed by $A$.
This feature of $\text{Set}$ is specific to that category, but I think it is instructive to see it in action. The finality of $1:A\to A$ is true in all $C/A$.
