definition. (co)unit [zhang2021type, 5.30] [tt-001V]

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Given an adjunction \(\mathscr {L} \dashv \mathscr {R}: {\cal C} \rightleftarrows {\cal D}\), the natural transformation \[\eta : \mathit {1}_{{\cal C}} \to \mathscr {L} \mathbin {\bullet } \mathscr {R} \] is called the unit of the adjunction, and \[\epsilon : \mathscr {R} \mathbin {\bullet } \mathscr {L} \to \mathit {1}_{{\cal D}}\] is called the counit.

We call an arrow \[\eta _X: X \to (\mathscr {L} \mathbin {\bullet } \mathscr {R})(X)\] a unit over \(X\), and \[\epsilon _Y: (\mathscr {R} \mathbin {\bullet } \mathscr {L})(Y) \to Y\] a counit over \(Y\). They are components of the natural transformations \(\eta \) and \(\epsilon \), respectively.

Diagramatically, the diagrams commute.