# Category theory/Natural transformation

## 1 Example: maybeToList

 map even $maybeToList$ Just 5

yields the same as

 maybeToList $map even$ Just 5

yields: both yield

 [False]

### 1.1 Vertical arrows: sides of objects

… showing the operation of the natural transformation.

$\eta : \Phi \to \Psi$
maybeToList :: Maybe a -> [a]

#### 1.1.1 Left: side of X object

 maybeToList :: Maybe Int -> [Int] Nothing [] Just 0 [0] Just 1 [1]

#### 1.1.2 Right: side of Y object

 maybeToList :: Maybe Bool -> [Bool] Nothing [] Just True [True] Just False [False]

### 1.2 Horizontal arrows: sides of functors

$f : X \to Y$
 even :: Int -> Bool

#### 1.2.1 Side of Φ functor

 map even:: Maybe Int -> Maybe Bool Nothing Nothing Just 0 Just True Just 1 Just False

#### 1.2.2 Side of Ψ functor

 map even:: [Int] -> [Bool] [] [] [0] [T]rue [1] [F]alse

### 1.3 Commutativity of the diagram

$\Psi(f) \cdot \eta_X = \eta_Y \cdot \Phi(f)$

both paths span between

$\Phi(X) \to \Psi(Y)$
 Maybe Int -> [Bool] map even . maybeToList maybeToList . map even Nothing [] [] Just 0 [True] [True] Just 1 [False] [False]

### 1.4 Remarks

• even
has a more general type (
Integral a => a -> Bool
) than described here
• Words “side”, “horizontal”, “vertical”, “left”, “right” serve here only to point to the discussed parts of a diagram, thus, they are not part of the scientific terminology.