  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 Brugui\`eres 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.
