except for (U).

-   Group algebras of finite noncommutative groups [@ambrose:hstar Example 4] satisfy everything except for (C).

-   Any nontrivial commutative (unital) Hopf algebra satisfies everything except for (F) by [@kock:frobenius Proposition 2.4.10].

It is worth noting that under additional assumptions, such as unitality and enrichment in abelian groups, (A) and (M) are known to imply (F) [@longoroberts:dimension Section 6].

We shall now *redefine* a Frobenius algebra[^3] in a dagger monoidal category to be an object $A$ equipped with a comultiplication $\delta : A \to A {\ensuremath{\otimes}}A$ satisfying (A), (C), (M) and (F). A Frobenius algebra which additionally has an arrow $\varepsilon \colon A \to I$ satisfying (U) will explicitly be called *unital*.

Regular representation as pointwise abstraction
-----------------------------------------------

As we have seen, unital Frobenius algebras allow us to define compact, and hence closed, structure. How much of this can we keep in key examples such as ${\ensuremath{\mathbf{Hilb}}}$?

The category ${\ensuremath{\mathbf{Hilb}}}$ has well-behaved duals, since $H \cong
H^{**}$, and indeed there is a conjugate-linear isomorphism $H \cong
H^{*}$. However, it is *not* the case that the tensor unit ${\ensuremath{\mathbb{C}}}$ is exponentiable in ${\ensuremath{\mathbf{Hilb}}}$, since if it was, we would have a bounded linear evaluation map $$H {\ensuremath{\otimes}}H^* \to {\ensuremath{\mathbb{C}}},$$ 