 Solution

The comma category $(X\downarrow G)$ has objects $(A,x)$ with $A\in \mathcal{C}$ and a function $x:X\to GA$. A morphism $(A,x)\to (B,y)$ is an arrow $f:A\to B$ such that $Gfx=y$. Identities and composition come from $\mathcal{C}$. When $X=1$, a map $1\to GA$ selects an element of $GA$, so this is the covariant category of elements of $G$.