Coend elements of a braided Hopf algebra

Let $H$ be a Hopf algebra in a braided rigid monoidal category $\mathcal{V}$ admitting a coend $C$. We define a ``coend element’’ of $H$ to be a morphism from $C$ to $H$. We then study certain coend elements of $H$, which generalize important elements (e.g., pivotal and ribbon elements) of a finite dimensional Hopf algebra over a field. This builds on prior work of Bruguières and Virelizier (2012) on $R$-matrices of braided Hopf algebras. As an application, we provide another description for pivotal and ribbon structures on the category ${\mathcal{V}}_H$ of $H$-modules.