Yoneda lemma for one object categories

Question

Let $G$ be a group and let $\mathbb{G}$ be the associated one object category. Is there an explicit presentation of representable functors from $\mathbb{G} \to $Set? If so how does the Yoneda lemma look like explicitly in this setting?

Answer 1

Let me write $\def\B{\mathbf{B}}\B G$ for what you call $\mathbb{G}$.

You can check that presheaves on $\B G$ are precisely the right $G$-sets: the unique point of $\B G$ is sent to some set $X$, and functoriality defines a group homomorphism $\def\op{\mathrm{op}}G^\op\to\operatorname{Aut}(X)$.

In particular, the (unique) representable functor corresponds to the $G$-set $G$ itself with the action given by right multiplication.

The Yoneda Lemma in this setting then says, for any $G$-set $X$, that elements of $X$ correspond naturally to $G$-equivariant maps $G\to X$: send $x\in X$ to the map $g\mapsto x.g$, and send a function $f:G\to X$ to $f(1_G)$.

Edit: to be a bit more explicit about the naturality, the group $G$ acts on both sides of this correspondence (the action on $X$ is given by the fact that $X$ is a $G$-set, and the action on $G$-equivariant functions $G\to X$ is componentwise), and the naturality of the Yoneda lemma says that this correspondence respects these $G$-actions.