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A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional. See for example this question for a discussion. Now every Hopf algebra $H$ admits a one-dimensional ...

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 ...