Groups as categories

Recall that to any group we can associate a category $B G$ that has a unique object, an arrow for each element of $G$ , and arrow composition given by the group law in $G$ . (See a bit more here.)

Show there is a correspondence^[1] between groups and one-object categories in which every arrow is invertible.
Suppose $G$ and $H$ are groups. Show that group morphisms $ϕ : G \to H$ correspond^[2] to functors $Φ : B G \to B H$ .
Recall that a permutation representation of $G$ is a group morphism $ϕ : G \to S_{X}$ , where $S_{X}$ is the permutation group on a set $X$ . Show that permutation representations of $G$ correspond to functors $B G \to S e t$ .
Suppose $ϕ, ψ : G \to H$ are group morphisms, with corresponding functors $Φ, Ψ : B G \to B H$ . Show there is a natural transformation $Φ \Rightarrow Ψ$ if and only if $ϕ$ and $ψ$ are conjugate, i.e., there is an element $h \in H$ such that $ψ (g) = h (ϕ (g)) h^{- 1}$ for all $g \in G$ .

You can turn this into a bijection if you are willing to talk about isomorphism classes of groups and the like. ↩︎
Following the previous footnote, you can turn this into an equivalence of certain categories. ↩︎