$\newcommand{\Hom}{\mathscr{Hom}}$ Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable;

Let $A$ be a commutative algebra in $C$, together with a morphism $p: A\to 1$ such that the composite $A\otimes A\overset\mu\to A\overset p\to 1$ is a perfect pairing, i.e. it induces an isomorphism $A\to \Hom(A,1)$, where $\Hom$ denotes the internal hom.

**Q1** Is it enough to conclude that $A$ has a structure of a Frobenius algebra in $C$ ? Where the comultiplication would be obtained by dualizing the multiplication ?

If I understand correctly the remark here, they seem to claim it's true in the category of vector spaces over a field - is it true more generally ? (this also seems to be definition 3.1 in the paper of Ross Street that is linked there, but because it talks about pseudomonoids in 2-categories etc. it's not clear that this really refers to the same thing)

My second question is the main one, I'm interested in it even if the answer to Q1 is no. I'm interested about the traces of $A$-modules. Indeed, it's not hard to show that an $A$-module which is dualizable is also dualizable as an object in $C$. In particular, it has a trace as an $A$-module (which is an element in $\hom_A(A,A) = \hom(1,A)$), and as an object of $C$ (an element in $\hom(1,1)$). My question is :

Q2: can these traces be compared in any way ? For instance, is it true that $Tr(A) p\circ Tr_A(x) \circ \eta = Tr(x)$ ? What about traces of $A$-module morphisms $x\to x$ ?

(where $\eta : 1\to A$ is the structure map, and $Tr_A$ is the trace as an $A$-module, $Tr$ the trace in $C$)

If a result holds under additional niceness assumptions, I'd be happy to hear about them too.

The reason I believe this isn't too far-fetched is that under these assumptions, $\Hom(A,1)\cong A$ as $A$-modules ($\Hom$ is the internal hom), and from that isomorphism it follows that $\Hom_A(x,A) \cong \Hom(x,1)$ for any $A$-module $x$, i.e. the dual is the same as an $A$-module and as an object in $C$. If you draw the "obvious" diagrams relating $Tr_A(x)$ and $Tr(x)$, some of them commute, and some others I'm not sure, but they seem to indicate that a relation as above could hold.

Moreover, here's a related paper, in which the authors show that a Frobenius functor preserves duality data in a sense, and they claim that $A\otimes - : C\to C$ is Frobenius when $A$ is a Frobenius ring. What I'm wondering is, in a sense, whether the forgetful $A-Mod \to C$ is also a Frobenius functor, so let me state this as a 3rd question, although I'm only interested in it insofar as it could help for Q2:

**Q3**: Is the forgetful functor $A-Mod\to C$ a Frobenius functor in the sense of the above paper ?