Contents

### Context

#### 2-Category theory

2-category theory

# Contents

## Definition

A pair

$(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$

of adjoint functors between categories $C$ and $D$, is characterized by a natural isomorphism

$C(L X,Y) \cong D(X,R Y)$

of hom-sets for objects $X\in D$ and $Y\in C$. Two morphisms $f:L X \to Y$ and $g : X \to R Y$ which correspond under this bijection are said to be adjuncts of each other. That is, $g$ is the (right-)adjunct of $f$, and $f$ is the (left-)adjunct of $g$. Sometimes one writes $g = f^\sharp$ and $f = g^\flat$, as in musical notation.

Sometimes people call $\tilde f$ the “adjoint” of $f$, and vice versa, but this is potentially confusing because it is the functors $F$ and $G$ which are adjoint. Other possible terms are conjugate, transpose, and mate.

## Properties

###### Proposition

Let $i_X : X \to R L X$ be the unit of the adjunction and $\eta_X : L R X \to X$ the counit.
• the adjunct of $f : X \to R Y$ in $D$ is the composite
$\tilde f : L X \stackrel{L f}{\to} L R Y \stackrel{\eta_Y}{\to} Y$
• the adjunct of $g : L X \to Y$ in $C$ is the composite
$\tilde g : X \stackrel{i_X}{\to} R L X \stackrel{R g}{\to} R Y$