Very general object-free categories? Is there a name for categories $\mathcal C$ such that $\text{Obj}(\mathcal C)$ coinside with $\text{Mor}(\mathcal C)$?
In the diagram below the morphisms are 
$a\overset{a}{\to}a$, $a\overset{b}{\to}c$, $d\overset{c}{\to}c$, 
$a\overset{d}{\to}e$, $e\overset{e}{\to}f$, $c\overset{f}{\to}e$.

I don't know if this is interesting mathematically. My idea is modelling. In the models all objects should be of the arrow-type in the same category. Nothing in the definition of categories exclude this. Domains and codomains can be defined as usual.

Depending on what precisely you are trying to model, you will need to specify some more axioms. You may then find out you are running into trouble. In any case, you may be interested in monoidal categories, or more generally in 2-categories. There is plenty online to read.
