are you fucking kidding

I'm probably completely overthinking this but anyway, if F : Set -> Ab is free and G : Ab -> Set is forgetful, there's a natural isomorphism Ab(Z,-) -> Set(*,G(-)):

ok comma categories have been bothering me a bit

I think it's mainly the definition I'm working with, where you define objects to be morphisms and morphisms to be pairs of morphisms that make commuting squares. but my issue is you can have the same pair of morphisms create different squares:

real

that's the first half of the iff at least, but it's also the better half because it contains this funky dance that you've probably seen before