Coproduct for a Frobenius algebra

Question

The definition of a Frobenius algebra given here describes it as a monoid and a comonoid in a monoidal category with a compatability condition. For the special case of the category of vector spaces a Frobenius algebra is then described as an unital algebra with a special type of linear form $\epsilon$. What I cannot see for this special case is how to produce a coproduct.

Answer 1

I believe you dualize the product $\mu:A\otimes A\to A$ to get $\mu^*:A^*\to A^*\otimes A^*$. By identifying $A$ with $A^*$ via $\epsilon$ you should get the coproduct $A\to A\otimes A$.