Adjunction

Formal Definition

Adjunction is a relationship only between Functors. Only functors can be Adjoint to each other.

• Let $\mathcal{A}$ and $\mathcal{B}$ be Categories
• Let $L$ and $R$ be Functors.
  • Left Adjoint (Functor)
    • $L:\mathcal{A}\to \mathcal{B}$
  • Right Adjoint (Functor)
    • $R:\mathcal{B}\to \mathcal{A}$
• Let $A\in \text{obj}(\mathcal{A})$
• Let $B\in \text{obj}(\mathcal{B})$

$${\text{hom}}_{\mathcal{B}}(L(A),B)={\text{hom}}_{\mathcal{A}}(A,R(B))$$