 Arrow category

The arrow category of $\mathcal{C}$ has morphisms $A_0\to A_1$ of $\mathcal{C}$ as objects and commutative squares as morphisms. It is the functor category $[2,\mathcal{C}]$, where $2$ is the category with one nonidentity arrow.

It is also the comma category $({\mathrm{id}}_{\mathcal{C}}\downarrow {\mathrm{id}}_{\mathcal{C}})$.