## Equivalent definitions of adjoint functors

- 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 φ
_{a,b}build a natural transformation A^{op}× B --> Set and that each φ_{a,b}is a bijection

- 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 --> 1
_{B}and - the counit η: 1
_{A}--> RL - with the triangle equations
- Rε • η
_{R}= 1, i.e. ∀ b ∈ B: Rε_{b}• η_{Rb}= 1_{b}, namely Rb --η_{Rb}--> RLRb --Rε_{b}--> Rb and - ε
_{L}• Lη = 1, i.e. ∀ a ∈ A: ε_{La}• Lη_{a}= 1_{a}, namely La --Lη_{a}--> LRLa--ε_{La}--> La

- Rε • η

- the unit ε: LR --> 1
- Given two categories A, B and a functor R: A <-- B and
- ∀a ∈ A there is an universal arrow η
_{a}: a --> Ru for R, - i.e. ∀ f: a --> Rc ∃! f': u --> c with f = η
_{a}• Rf'

- ∀a ∈ A there is an universal arrow η
- the dual with a functor L: A --> B and couniversal arrows

Proof: 1 ⇒ 2

- define ε
_{b}= φ^{-1}(1_{Rb}) - the diagram shows, that this defines a morphism ε
_{b}: RLb --> b and that it is natural - (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)
- dually η
_{a}= φ(1_{La}) is a natural transformation

- This diagram prooves one of the dual triangle equations.

Proof: 2 ⇒ 3

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

Proof: 3 ⇒ 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 (η
_{a'}f)' of the universal arrow a --η_{a}--> La for η_{a'}f

- we define φ
_{a, b}: [La, b] --> [a, Rb] as f' |--> Rf' η_{a}

- φ
_{a, b}is bijective because of the ∃! in the universal arrow - the diagram beside spells out naturality of φ. But by construction of L we have η
_{a}α = RLα η_{a'}. Thus, the diagram is commuatitve, and hence, φ natural. - 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..