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 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.

On the other hand, a Hopf algebra $H$ 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 spells it out. I'm sure I could figure this out myself eventually and type up an nice answer, but I'll bet someone on this site already knows, and can provide some useful insight too.