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.