What is the name for this generalization of a category?

Question

Is there a name for an object which is made of a set of objects, and a set of arrows which can be from objects/arrows to objects/arrows (all four combinations)?

Equivalently, this is a category where each arrow is identified with a unique object in the category.

Answer 1

It echoes with the generalized categories of Lucius Schoenbaum. Basically it takes the unisorted presentation of category theory and drops the requirement that $ss=st=s$ and $tt=ts=t$ (where $s$ is the source map and $t$ the target map).

Answer 2

Let $\mathcal{C}$ be a category. Let $\text{Arr}(\mathcal{C})$ be the category of morphisms of $\mathcal{C}$. Perhaps what you want is simply a category $\mathcal{C}$ equipped with a "nice" functor $F: \text{Arr}(\mathcal{C}) \to \mathcal{C}$. Then, for instance, if $f:A\to B$ and $g:B\to C$ are morphisms in $\mathcal{C}$, then $\alpha : F(f)\to F(g)$ is a morphism between morphisms.