How does the functor $F: \textbf{C}/C \to \textbf{C}$ "forget about the base object" $C$? Let $\textbf{C}/C$ be a slice category with base object $C$.
The functor that comes to mind is the one such that objects $f:X \to C$ are mapped to $X$, and arrows $a: X \to X'$ are mapped to themselves.
Let $F$ be this mapping.  Then since, the identiy on the object $f: X\to C$, is $1_X$, we have that $F(1_f) = F(1_X) = 1_{F(X)}$. Further, if $a: f \to g$, then define $F(a) = a$, for which we have $F(a): F(f) \to F(g)$, and finally $F(i \circ j) = i \circ j$.  
First of all how can I make it explicit that $a : f \to g$ in $\textbf{C}/C$,  but in $\textbf{C}$, $a: \text{dom}(f) \to \text{dom}(g)$?
And then how is $C$ "forgotten"?
 A: $a : f \to g$ is already an explicit way to say $a : f \to g$. If you want to put the category in the notation, a simple way is to use a homset; two notations for such are
$$ a \in \hom_{\mathcal{C}/C}(f, g) \qquad \qquad \text{or} \qquad \qquad
a \in (\mathcal{C}/C)(f, g) $$
Alternatively, rather than writing $f$ for an object of $\mathcal{C}/C$, you may instead write $(X,f)$ for the object. Or even the whole diagram $X \xrightarrow{f} C$. Then maybe having both $a : (X,f) \to (Y,g)$ and $a : X \to Y$ is already sufficient for your tastes.
As for forgetting $C$, it's just what it sounds like; applying $F$ just discards everything related to $C$. For example, on the left is a depiction of a commutative triangle in $\mathcal{C} / C$. Applying $F$ just lops off everything crossing the dotted line, to give the triangle on the right.

A: If $f:X\rightarrow C$ and $g:Y\rightarrow C$, $a:f\rightarrow g$ is a morphism $a:X\rightarrow Y$ such that $f=g\circ a$.
A: All you need to do is say what $F$ does to objects and morphisms, and then check that it is a functor.  An object in $C/C$ is of the form $a:X\to C$ and $F$ sends it to $X$. A morphism from $a:X\to C$ to $\ b:Y\to C$ is a morphism $\phi:X\to Y$ such that $b\circ \phi =a$. Now, $F$ simply sends $\phi$  to $\phi,\ $and forgets the commuting triangle. Of course, you need to check associativity and the fact that $F1_X=1_{FX},\ $but these are pretty obvious.
Actually, there are several different "forgetful" functors from slice categories. See here.
