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