TITLE: Categorical meaning of the action of a group on itself by conjugation
QUESTION [6 upvotes]: For a group $G$ there are always two actions of $G$ on itself: $g*h=gh$ and $g\star h=ghg^{-1}$. As objects in the category $G$-$Set$, the former is the free $G$-set on a singleton set. When $G$ is considered as a one-object category, this same $G$-action is also the unique representable functor on $G$. Is there any similar way to understand the action by conjugation in categorical terms?

REPLY [2 votes]: Associated to any action of a group $G$ on a set $X$ is its action groupoid, which you can think of as the homotopy quotient of $X$ by the action of $G$, which I'll write $X//G$. The unique map $X \to \bullet$ induces a map 
$$X//G \to \bullet//G$$
where $\bullet//G$, otherwise known as $BG$, is the groupoid with one object with automorphism group $G$. This construction extends to an equivalence of categories between the category of $G$-sets and the category of groupoids equipped with a covering map to $BG$ in a suitable sense. 
So, to understand the conjugation action of $G$ on itself we can equivalently attempt to understand the action groupoid $G//G$, together with its map to $BG$. The action groupoid $G//G$ can in turn be understood as the free loop space
$$G//G \cong L(BG) \cong [S^1, BG] \cong [B\mathbb{Z}, BG]$$
of $BG$, and the map to $BG$ can be understood as induced by applying $[-, BG]$ to any map $\bullet \to S^1$. 
Note that $G$ acting on itself by left multiplication appears as the point $\bullet$, together with any map $\bullet \to BG$.