Let $\mathcal{V}$ be a closed braided monoidal category and $\mathcal{V}-Cat$ the monoidal bicategory of small $\mathcal{V}$-enriched categories. Let $\mathcal{C}$ be a pseudo-comonoid in $\mathcal{V}-Cat$, which can be seen as the dual 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 on $\mathcal{C}$, so that the $\mathcal{V}$-functor category $[\mathcal{C},\mathcal{V}]$ becomes closed monoidal.
In [Day, Exp. 5.1] it is said that that if $\mathcal{C}$ has a single object $\bullet$, then $\mathcal{C}(\bullet,\bullet)$ is a Hopf monoid.
I'm completely fine with the fact that it is a bimonoid, but where does the antipode come from?
A (supposed to be) counterexample:
Let's start with a bimonoid $B$ in $\mathcal{V}$ and set $\mathcal{C}(\bullet,\bullet)=B$. I'm pretty sure that the comonoid structure induces a pseudo-comonoid structure on $\mathcal{C}$. Thus, by [Day, Exp. 5.1], $B$ is a Hopf monoid, so every bimonoid is a Hopf monoid?!
So, if this is 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 it 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 - On closed categories of functors
