[Verse 1]
When you build an object, forget the blueprints inside
Universal properties tell you how it talks outside
The product A times B, with arrows π one and two
Any pair of maps will factor through, uniquely through
[Chorus]
Universal means the BEST
Unique up to isomorphism test
Don't construct it, characterize
By what it does, not what it hides
The tensor product, kernel too
All defined by what they do
[Verse 2]
Free group on a set is most general you can make
Bilinear maps through tensor, that's the handshake
Philosophy runs deeper than the surface that you see
This object is the champion of its property
[Chorus]
Universal means the BEST
Unique up to isomorphism test
Don't construct it, characterize
By what it does, not what it hides
The tensor product, kernel too
All defined by what they do
[Verse 3]
Adjunctions are negotiations, perfect harmony
F goes left and G goes right, translators fluently
Hom D of F A comma B matches Hom C naturally
A comma G of B, that's the key identity
[Bridge]
Free adjoint to forgetful, product with internal hom
Tensor dancing with the hom functor, sigma delta pi
Left approximates optimally, right gives you the bound
Most constructions that matter are adjoints that you've found
[Chorus]
Universal means the BEST
Unique up to isomorphism test
Adjunctions negotiate
Between categories that translate
F and G are speaking true
Perfect compatibility through
[Outro]
Don't ask what it's made of, ask what does it do
Universal properties and adjunctions guide you through