What properties of a finite group fibre functor give its endomorphisms a hopf algebra structure?

Question

Tannaka duality for a finite group lets us recover the group algebra $\mathbb{C}[G]$ as the endomorphisms of the forgetful functor $F:RepG\rightarrow Vect$, and taking the monoidal automorphisms recovers the grouplike elements of this hopf algebra, which we can recognise as just our group $G$.

Is there a diagrammatic way of getting the comultiplication/antipode structure of this hopf algebra?

By diagrammatic, I mean a description that shouldn't rely on the objects of our categories, and is expressible in the higher level structure, such as the multiplication being composition of endomorphisms.

I would think such a description would apply for arbitrary functors between categories that share enough of the nice properties of $RepG$ and $Vec$, but as I am interested in $RepG$, an ad-hoc method for this fibre functor would also be of interest.

Answer 1

The tensor product functor $\otimes : \text{Rep}(G) \times \text{Rep}(G) \to \text{Rep}(G)$ is "bilinear": it preserves colimits in both variables. It can therefore be regarded as being a "linear" functor on a "tensor product" category

$$\otimes : \text{Rep}(G) \otimes \text{Rep}(G) \to \text{Rep}(G)$$

which can be identified with $\text{Rep}(G \times G)$. In general we have $\text{Mod}(R) \otimes \text{Mod}(S) \cong \text{Mod}(R \otimes_k S)$ for $k$-algebras $R$ and $S$, and we have $k[G] \otimes_k k[G] \cong k[G \times G]$. All of this can be understood rigorously using either the Morita 2-category of bimodules or an enriched version of the universal property of presheaves, but if you can take it on faith that this sort of thing makes sense at least when the representation category is semisimple we don't have to go into it.

Composing $\otimes$ with the fiber functor $F : \text{Rep}(G) \to \text{Vect}$ then gives a composite functor

$$F \circ \otimes : \text{Rep}(G \times G) \to \text{Vect}.$$

The endomorphisms of this functor give $k[G \times G]$, and whiskering along $\otimes$ should define a map $k[G] \to k[G] \otimes_k k[G]$ reproducing the comultiplication (although I haven't checked this).

I think we can also construct the antipode but the idea I have in mind involves some funny business with the dualization functor.