I am working with a rigid, abelian braided category $\mathcal{C}$, which has a natural Hopf algebra object defined via the coend $C=\int^{X \in \mathcal{C}}X^{*}\otimes X$, where $X^{*}$ is the dual object to $X$. The coend exists since the category $\mathcal{C}$ is bounded.
We know precisely what the coend of the category $\mathcal{C}$ is, including its structure morphisms as a Hopf algebra object. Moreover, we have a non-degenerate pairing $\omega: C \otimes C \rightarrow \mathbb{1}$, which we can also describe precisely. We know that the existence of a Hopf algebra object implies the existence of a Frobenius algebra object.
The question that concerns me is the following: is it possible to construct the Frobenius algebra object out of this data? If yes, could you please give a reference regarding that construction or something similar to it. I appreciate any comments or help. Thank you in advance!