Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 25 / 5 / a / Solution

The comma category $(X\downarrow G)$ has objects $(A,x)$ with $A\in \mathcal{C}$ and a function $x:X\to G(A)$. Its morphisms $(A,x)\to (B,y)$ are maps $f:A\to B$ in $\mathcal{C}$ satisfying $G(f)x=y$. The identity morphism and composition in a category are inherited from $\mathcal{C}$; the functor laws make the defining equations stable under both.

For a singleton $1$, its map $1\to G(A)$ is just an element $a\in G(A)$. Thus $(1\downarrow G)$ is the covariant category of elements of $G$, with arrows $f:(A,a)\to (B,b)$ satisfying $G(f)(a)=b$.