# arrow category

## Primary tabs

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Let $\mathcal{C}$ be a category.  The \emph{arrow category} of $\mathcal{C}$ is the functor category $\mathcal{C}^\textbf{2}$.  Here, $\textbf{2}$ is the ordinal category consisting of $0,1$.  Specifically, the contents of $\mathcal{C}^\textbf{2}$ are:
\begin{enumerate}
\item an object of $\mathcal{C}^\textbf{2}$ is an arrow (morphism) of $\mathcal{C}$
\item given two objects of $\mathcal{C}^\textbf{2}$, say $A \stackrel{f}{\longrightarrow} B$ and $A' \stackrel{g}{\longrightarrow} B'$, a morphism (of $\mathcal{C}^\textbf{2}$) 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.
\end{enumerate}
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