What is the universal property of the tangent bundle of a smooth manifold?

The process of writing my own notes on smooth manifolds have led me to wonder about this. All I've really found is the following:

In addition to Madame Ehresmann's references, there is in Spivak's Comprehensive Introduction... an abstract characterization of the tangent bundle ( removed from the main text in the second edition `due to the pressure of public distaste')

Taken from http://permalink.gmane.org/gmane.science.mathematics.categories/1189

I have neither Madame Ehresmann's notes, nor an older edition of Spivak.

Sincerely, Eivind

Solution 1:

The material in question was never removed from the textbook. It was only removed from the "main text": it was taken out of the central discussion of the tangent bundle, and put in its own section.

The "public distaste" remark comes from the front matter of the second edition, where the author writes

I bowed to the pressure of public distaste and removed the characterization of the tangent bundle up to natural equivalence from the body of Chapter 1.3, and put it instead into an Addendum.

In the most recent edition, this addendum is at the end of Chapter 3, in a section called "Addendum: Equivalence of Tangent Bundles."