and hence its adjoint ${\ensuremath{\mathbb{C}}} \to H {\ensuremath{\otimes}}H^*$, and a compact structure.

We shall now present an axiom which captures what seems to be the best we can do in general in the way of a ‘transfer of variables’. It is, indeed, a general form, meaningful in any monoidal dagger category, of a salient structure in functional analysis.

Suppose we have a comultiplication $\delta \colon A
\to A {\ensuremath{\otimes}}A$, and hence a multiplication $\mu = \delta^\dag
\colon A {\ensuremath{\otimes}}A \to A$. We can *curry* the multiplication (this process is also called $\lambda$-abstraction [@barendregt:lambdacalculus]) for *points*—this is just the regular representation![^4] Thus we have a function $R \colon {\ensuremath{\mathbf{D}}}(I,A) \to {\ensuremath{\mathbf{D}}}(A,A)$ defined by $$R(a) = \mu {\circ}({\ensuremath{\mathrm{id}_{}}}{\ensuremath{\otimes}}a) = {
 \begin{tikzpicture}[scale=0.5]  \PandocStartInclude{tikz/regular.tikz}\PandocEndInclude{input}{24}{23} \end{tikzpicture}}.$$ If $\mu$ is associative, this is a semigroup homomorphism.

Axiom (H)
---------

An endomorphism homset ${\ensuremath{\mathbf{D}}}(A,A)$ in a dagger category ${\ensuremath{\mathbf{D}}}$ is not just a monoid, but a *monoid with involution*, because of the dagger. We say that $(A, \mu)$ *satisfies axiom (H)* if there is an operation $a
\mapsto a^*$ on ${\ensuremath{\mathbf{D}}}(I, A)$ such that $R$ becomes a homomorphism of involutive semigroups, [*i.e.* ]{} $$R(a^*) = R(a)^\dag$$ for every $a \colon I \to A$. This unfolds to $$\begin{aligned}
    