Category theory/Natural transformation

Example: maybeToList

 map even $ maybeToList $ Just 5

yields the same as

 maybeToList $ map even $ Just 5

yields: both yield

 [False]

Vertical arrows

$\eta :\Phi \to \Psi$

maybeToList :: Maybe a -> [a]

$\eta _{X}:\Phi (X)\to \Psi (X)$

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

$\eta _{Y}:\Phi (Y)\to \Psi (Y)$

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

Horizontal arrows

/Note: even has a more general type (Integral a => a -> Bool) than described here/

$f:X\to Y$

 even :: Int -> Bool

$\Phi (f):\Phi (X)\to \Phi (Y)$

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

$\Psi (f):\Psi (X)\to \Psi (Y)$

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

Commutativity of diagram

$\eta _{Y}\cdot \Phi (f)=\Psi (f)\cdot \eta _{X}$

	map even . maybeToList	maybeToList . map even
Nothing	[]	[]
Just 0	[True]	[True]
Just 1	[False]	[False]