Let $\mathcal{C}$ be a category. The arrow category of $\mathcal{C}$ is the functor category $\mathcal{C}^{2}$ Here, ${2}$ is the ordinal category consisting of $0,1$ Specifically, the contents of $\mathcal{C}^{2}$ are:

1. an object of $\mathcal{C}^{2}$ is an arrow (morphism) of $\mathcal{C}$
2. given two objects of $\mathcal{C}^{2}$ say $A \stackrel{f}{\longrightarrow} B$ and $A' \stackrel{g}{\longrightarrow} B'$ a morphism (of $\mathcal{C}^{2}$ from $f$ to $g$ consists of an ordered pair $(h,k)$ where $A \stackrel{h}{\longrightarrow} A'$ and $B \stackrel{k}{\longrightarrow} B'$ such that the following diagram $$\xymatrix{ A \ar[r]^h \ar[d]_{f} & A' \ar[d]^g \\ B \ar[r]_k & B'} $$ is a commutative diagram.