nLab
unit of an adjunction

Definition

Given an adjunction

$$(L \dashv R) \;\colon\; X \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}} Y$$

there is a natural transformation (or more generally, a $2$-morphism) $\eta\colon id_X \to R \circ L$, called the unit of the adjunction. (This is so called because $R \circ L$ is a monad, which is a kind of monoid object, and $\eta$ is the identity of this monoid. Since ‘identity’ in this context would suggest an identity natural transformation, we use the synonym ‘unit’.)

Similarly, there is $2$-morphism $\epsilon\colon L \circ R \to id_Y$, called the counit of the adjunction. (This is the co-identity of the comonad $L \circ R$.)

Properties

General

Unit and counit of an adjunction satisfy the triangle identities.

An adjunct is given by precomposition with a unit or postcomposition with a counit.

Relation to monads

Every adjunction $(L \dashv R)$ gives rise to a monad $T \coloneqq R \circ L$. The unit of this monad $id \to T$ is the unit of the adjunction, $id \to R \circ L$.

Related concepts

• adjunction

• zig-zag law/triangle identity

• unit

  • unit object

  • unit of an adjunction

  • unit of a monad

• adjunct