# nLab unit of a monad

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Defintion

Given a monad $T \;\colon\; \mathcal{C} \to \mathcal{C}$, its unit is the natural transformation

$\epsilon \;\colon\; id_{\mathcal{C}} \to T$

which is part of the definition of monad. Hence for every object $X \in \mathcal{C}$ the component of the unit on $X$ is a morphism

$\epsilon_X \;\colon\; X \to T(X)$

in $\mathcal{C}$.

Dually, there is a counit of a comonad.

## Properties

If $(L \dashv R)$ is an adjunction that gives rise to the monad $T$ as $T \simeq R \circ L$, then the unit of the monad is equivalently the unit of the adjunction.