Adjunction is the “**weakening**of **Equality”** in Category Theory.

**Equivalence** of 2 Categories if:

5.2 Adjunction definition: $latex (L, R, epsilon, eta )$ such that the 2 **triangle identities** below ( red and blue) exist.

6.1**Prove**: Let C any category, D a Set.

$latex boxed {text {C(L 1, -)} simeq text {R}}&fg=aa0000&s=3$

$latex {text {Right Adjoint R in Set category is }}&fg=aa0000&s=3$ $latex {text {ALWAYS Representable}}&fg=aa0000&s=3$

1 = Singleton in Set D

From an element in the singleton Set always ‘picks’ (via the function) an image element from the other Set, hence :

$latex boxed {text{Set (1, R c) } simeq text{Rc }}&fg=0000aa&s=3$

**Examples** : Product & Exponential are Right Adjoints

**Note**: Adjoint is a **more powerful** concept to understand than the universal construction of Product and Exponential.