TITLE: groups as categories and their natural transformations
QUESTION [6 upvotes]: If one views a group as a one object category with the elements of the group as morphisms then a natural transformation between functors of such categories is an inner automorphism, i.e. if we have two group homomorphisms $f,g: A\to B$ then a natural transformation $\eta :f\to g$ is just an element $b\in B$ such that $f(a)\cdot b = b \cdot g(a)$ which can be rewritten as $f(a)=b \cdot g(a)\cdot b^{-1}$. This isn't the only way to turn groups into categories. Another way is to take the elements of the group as objects and to have a morphism $h_a:a\to b$ if $h\cdot a=b$. If we view groups in this way then are the natural transformations again something nice like inner automorphisms?

REPLY [4 votes]: I like the notation $\mathcal{B}G$ and $\mathcal{E}G$ for the two constructions of a category out of a group $G$ in David's question. $\mathcal{E}G$ is what Tom calls the codiscrete category $C(G)$. 
Of course there is a third construction: it has $G$ as the objects, and only identity morphisms. Let's denote this category again by $G$.
The notation is nice because you can take the nerve of any category $\mathcal{C}$, and then geometrically realize. If we denote the resulting space by $|\mathcal{C}|$, 

$|\mathcal{B}G|$ is a classifying space for $G$.
$|\mathcal{E}G|$ is a universal principal $G$-bundle  
$|G|$ is just $G$.