Comma Category

Let $\mathcal{A},\mathcal{B},\mathcal{C}$ be categories, and let $F:\mathcal{A}\Rightarrow \mathcal{C}$ and $G:\mathcal{B}\Rightarrow \mathcal{C}$ be functors.

The comma category, written $(F\downarrow G)$, is the category:

• with objects $\Sigma a:\mathcal{A},b:\mathcal{B}.\mathcal{C}[F(a),G(b)]$
• with morphisms from $(a_1,b_1,f_1:F(a_1)\to G(b_1))$ to $(a_2,b_2,f_2:F(a_2)\to G(b_2))$ given by pairs $(\alpha :a_1\to a_2,\beta :b_1\to b_2)$ such that $G(\beta )\circ f_1=f_2\circ F(\alpha )$.