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