# Categories: Adjoint Functors

## Equivalent definitions of adjoint functors

1. Given two categories A, B and two functors L: A --> B and R: A <-- B. L is left adjoint to R iff there is a natural isomorphism
• φ: R[L _, _ ] --> L[ _, R _ ]
• natural isomorphism means that the φa,b build a natural transformation Aop × B --> Set and that each φa,b is a bijection
2. Given two categories A, B and two functors L: A --> B and R: A <-- B. L is left adjoint to R iff there are natural transformations
• the unit ε: LR --> 1B and
• the counit η: 1A --> RL
• with the triangle equations
• Rε • ηR = 1, i.e. ∀ b ∈ B: Rεb • ηRb = 1b, namely Rb --ηRb--> RLRb --Rεb--> Rb and
• εL • Lη = 1, i.e. ∀ a ∈ A: εLa • Lηa = 1a, namely La --Lηa --> LRLa--εLa--> La
3. Given two categories A, B and a functor R: A <-- B and
• ∀a ∈ A there is an universal arrow ηa: a --> Ru for R,
• i.e. ∀ f: a --> Rc ∃! f': u --> c with f = ηa • Rf'
4. the dual with a functor L: A --> B and couniversal arrows

Proof: 1 ⇒ 2

1. define εb = φ-1(1Rb)
2. the diagram shows, that this defines a morphism εb: RLb --> b and that it is natural
3. (furhtermore, the equation in the green box shows that φ is defineded by its values on 1, this is the contents of the yoneda lemma)
4. dually ηa = φ(1La) is a natural transformation
5. This diagram prooves one of the dual triangle equations. Proof: 2 ⇒ 3

1. Given a ∈ A we use a --ηa--> RLa as universal arrow.
2. For any f: a --> Rc we find the required f' = εc Lf : La --> c as shown in the upper diagram
3. On the other hand, the lower diagram shows that each such f' can be calculated from Rf' ηa which is nothing but the original f. Thus it is unique.

Proof: 3 ⇒ 1

1. we construct L as
• the image La of an object a ∈ A is u from the universal arrow a --> Ru
• the image Lf of a morphism f: a --> a' is the unique (ηa' f)' of the universal arrow a --ηa--> La for ηa' f
2. we define φa, b: [La, b] --> [a, Rb] as f' |--> Rf' ηa
3. φa, b is bijective because of the ∃! in the universal arrow
4. the diagram beside spells out naturality of φ. But by construction of L we have ηa α = RLα ηa'. Thus, the diagram is commuatitve, and hence, φ natural.
5. thus, φ is the required natural isomorphism [L_, _] --> [_, R_] for the adjunction

Proof: ⇐ 4 ⇒

• we omit these proofs because 3 and 4 are dual, left and right adjoint are dual, etc..