Arrow Category

The arrow category $\mathcal{C}^{\to}$ is the category where

• Objects are triples $X:\mathcal{C},Y:\mathcal{C},f:{{X}\to{Y}}$
• Morphisms between triples $(A,B,f)$ and $(X,Y,g)$ are commuting squares

Note that the arrow category of $\mathcal{C}$ is the Comma Category $mathrm{Id}{ifx..lse\par i}/mathrm{Id}{ifx..lse\par i}$. Equivalently, the arrow category is the Functor Category $[2,\mathcal{C}]$, where $2$ is the Walking Arrow category.

Properties

• The arrow category is a functor category, so we can compute Limits and Colimits pointwise. For instance, the product of $(A,B,f)$ and $(X,Y,g)$ is given by $(A\times X,B\times Y,f\times g)$.