For categories $\mathcal{C}$ and $\mathcal{D}$, a pair of functors $\mathcal{C} \xrightarrow{\;L\;} \mathcal{D}$ and $\mathcal{C} \xleftarrow{\;R\;} \mathcal{D}\,$ are an [adjoint pair](https://ncatlab.org/nlab/show/adjunction) if we have natural transformations $\eta \colon 1_\mathcal{C} \Rightarrow RL$ and $\varepsilon \colon LR \Rightarrow 1_\mathcal{D}$, called the *unit* and *counit* respectively, that satisfy the [triangle identities](https://ncatlab.org/nlab/show/triangle+identities).

On the other hand, a [Hopf algebra](https://ncatlab.org/nlab/show/Hopf+algebra#definition) $H$ is an associative and coassociative bialgebra equipped with a multiplication $\nabla$, a comultiplication $\Delta$, an antipode $S$, and *unit* and *counit* maps $\eta$ and $\varepsilon$ such that $\nabla \circ (1_H \otimes S) \circ \Delta = \Delta \circ (S \otimes 1_H) \circ \nabla = \eta\circ \varepsilon$.

How are these two different definitions of *unit*/*counit* related? I mean, I know they must be the same idea when viewed in the right context, but I haven't figured it out yet, and neither nLab nor [Wikipedia](https://en.wikipedia.org/wiki/Hopf_algebra#Formal_definition) spells it out. I'm sure I could figure this out myself eventually and type up an nice answer (and will if no one else feels like typing it up), but I'll bet someone on this site already knows the relationship well and can provide some useful insight too.