[Verse 1]
Between two functors there's a bridge to build
A systematic way that maps unfold
For every object in the source category
A morphism waits to tell its story
From F of A to G of A it flows
But there's a rule that every mapper knows
[Chorus]
Natural transformation, commutes the square
Eta A to eta B, the pattern's always there
F composed with eta equals eta composed with G
Naturality condition sets the morphisms free
No arbitrary choices, works the same way through
Double dual isomorphism shows what natural can do
[Verse 2]
Take a vector space and map it twice
To its dual and back, the construction's nice
Send vector v to the evaluation map
Phi goes to phi of v, close the gap
No basis needed, choice-independent gleam
This transformation's natural, not just a scheme
[Chorus]
Natural transformation, commutes the square
Eta A to eta B, the pattern's always there
F composed with eta equals eta composed with G
Naturality condition sets the morphisms free
No arbitrary choices, works the same way through
Double dual isomorphism shows what natural can do
[Bridge]
When the diagram commutes for every single map
You've found a natural way to close the gap
Systematic relationships between the functors' dance
Not just one object's isolated circumstance
[Verse 3]
Three examples illuminate the concept clear
First the double dual we've already seen here
Second, take identity to any functor F
The natural inclusion makes the structure deaf
Third, the determinant from matrices to scalars
Natural transformation, one of category's scholars
[Chorus]
Natural transformation, commutes the square
Eta A to eta B, the pattern's always there
F composed with eta equals eta composed with G
Naturality condition sets the morphisms free
No arbitrary choices, works the same way through
Double dual isomorphism shows what natural can do
[Outro]
Independence from choices, that's the key
Mathematical naturality sets us free