sense we will now make precise. First, we make Frobenius algebras and [H\*]{}-algebras into categories. While other choices of morphisms can fruitfully be made [@heunen:copyables], the following one suits our current purposes.

\[def:morphisms\] Let ${\ensuremath{\mathbf{D}}}$ be a symmetric monoidal dagger category. We denote by ${\ensuremath{\mathbf{HStar}}}({\ensuremath{\mathbf{D}}})$ the category whose objects are [H\*]{}-algebras in ${\ensuremath{\mathbf{D}}}$, and by ${\ensuremath{\mathbf{Frob}}}({\ensuremath{\mathbf{D}}})$ the category whose objects are Frobenius algebras in ${\ensuremath{\mathbf{D}}}$. A morphism $(A,\delta) \to (A',\delta')$ in both categories is a morphism $f \colon A \to A'$ in ${\ensuremath{\mathbf{D}}}$ satisfying $(f {\ensuremath{\otimes}}f) {\circ}\delta = \delta' {\circ}f$ and $f^\dag {\circ}f = {\ensuremath{\mathrm{id}_{}}}$.

Every object in ${\ensuremath{\mathbf{PInj}}}$ carries a unique H\*-algebra structure, namely $\delta(a) = (a,a)$.

Let $\delta = (\xymatrix@1{A & \;D\; 
  \ar@{ >->}|-{\delta_1}[l] \ar@{ >->}|-{\delta_2}[r] & A \times A})$ be an object of ${\ensuremath{\mathbf{HStar}}}({\ensuremath{\mathbf{PInj}}})$. Because of (M), we may assume that $\delta_1 = {\ensuremath{\mathrm{id}_{}}}$. By (C), we find that $\delta_2$ is a tuple of some $d \colon A \to A$ with itself. It follows from (A) that $d = d {\circ}d$. Finally, since ${\ensuremath{\mathbf{PInj}}}$ is monoidally well-pointed, $\delta$ satisfies (F) by Lemma \[lem:HAWPimplyF\]. Writing out what (F) means gives $$\{((d(b),b),(b,d(b))) \mid b \in A \} 
     = \{((c,d(c)),(d(c),c)) \mid c \in A \}.$$ Hence for all $b \in 