$\newcommand{Hom}{\operatorname{Hom}}$
Consider a category $\mathbf{C}$, and draw a graph according to the following rules:
- Place all objects onto the graph
- Between every two objects $A,B\in\operatorname{obj}(\mathbf{C})$, if $\Hom(A,B)\neq\emptyset$, draw an arrow between $A, B$. Take that arrow to represent the entire hom-set of morphisms between $A$ and $B$.
- Now take the line-graph of the above graph and use it to define a category $\mathbf{C}^\prime$, whose objects are collections of morphisms in $\mathbf{C}$.
Perhaps more clearly, suppose that commutative squares like the following occur in $\mathbf{C}$:
$$ \require{AMScd} \begin{CD} A @>{f}>> A^\prime\\ @VV{h}V @VV{h^\prime}V \\ B @>{f^\prime}>> B^\prime \end{CD}$$
Then we take the entire hom-sets $\Hom(A,B), \Hom(A^\prime,B^\prime)$ to be objects $H, H^\prime$ respectively in $\mathbf{C^\prime}$. Furthermore, $F=\langle f, f^\prime\rangle$ is a morphism in $\mathbf{C^\prime}$; in general, a morphism $\langle f, f^\prime\rangle$ exists in $\Hom(H,H^\prime)$ iff there exists a pair of arrows $h\in H, h^\prime\in H^\prime$ so as to make the above diagram commute. In that regard, $F\circ G$ is meaningful iff the targets of $f,f^\prime$ are respectively the domains of $g,g^\prime$; in other words if both pairs share an object each. In that sense the objects of $\mathbf{C}$ define the morphisms of $\mathbf{C^\prime}$, thus the identification with the line graph of the diagram of $\mathbf{C}$.
In particular and for example, if $\mathbf{C}$ is preadditive, then all objects in $\mathbf{C^\prime}$ are additive abelian groups, and the morphisms $\mathbf{C^\prime}$ are all group homomorphisms.
Now, it's entirely possible I'm misunderstanding what comma categories are; but as I understand it, the category $(\mathbf{C}\downarrow\mathbf{C})$ has as its objects individual morphisms of $\mathbf{C}$, which isn't quite what I'm looking for. Am I wrong? If not, is there an existing name for categories as I've described them above?