# ct.category theory – Hopf monoid from comonoidal structures

Let $$mathcal {V}$$ to be a closed braided monoidal category and $$mathcal {V} -Cat$$ the monoidal bicategory of small $$mathcal {V}$$enriched categories. Let $$mathcal {C}$$ to be a pseudo-comonoid $$mathcal {V} -Cat$$, which can be seen as the double notion of a small monoidal $$mathcal {V}$$enriched category. In [Day, Ch.5] it is shown that from this we can build a promonoidal structure $$mathcal {C}$$, so that the $$mathcal {V}$$-factor category $$[mathcal{C},mathcal{V}]$$ becomes monoidal closed.

In [Day, Exp. 5.1] we say that if $$mathcal {C}$$ has a single object $$bullet$$then $$mathcal {C} ( bullet, bullet)$$ is a Hopf monoid.

I totally agree with the fact that it's a bimonoid, but where does the antipode come from?

One (supposed to be) counterexample:

Let's start with a bimonoid $$B$$ in $$mathcal {V}$$ And put $$mathcal {C} ( bullet, bullet) = B$$. I'm pretty sure that the comonoid structure induces a pseudo-comonoid structure on $$mathcal {C}$$. So, by [Day, Exp. 5.1], $$B$$ is a Hopf monoid, so each bimoide is a Hopf monoid ?!

So, if it's really a mistake, what kind of structure or property do we need for $$mathcal {C}$$ (in addition to being a pseudo-comonoid), so that if he has a single object, then (a) $$H: = mathcal {C} ( bullet, bullet)$$ is a Hopf monoid in $$mathcal {V}$$ and B) $$[mathcal{C},mathcal{V}]$$ is the usual closed monoidal category of $$H$$-modules?

[Day] Brian Day – Categories of closed functors