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') Kirill Mackenzie

Given an object M in the ``normal'' category of finitely dimensional smooth manifolds Man (not in SDG sense), what it the universal property of the tangent bundle TM?

So far, I found only the following:

For every manifold M there is a functor F:I -> Man0, where Man0 is category of open areas in R^n and smooth mapping, such that M=Colim F, F corresponding to the atlas on M and M is represented as a result of gluing instances of R^n in the atlas. This functor can be trivially modified (by multiplying its values on objects on R^n and modifying morphisms appropriately) to get functor TF:I -> Man0, such that TM=Colim TF.

But this doesn't seem satisfactory because:

1. Construction of TF follows one particular construction of TM as a set of triples (x,(U,f),h) where x \in U, (U,f) is in atlas and h \in R^n with appropriate points identified.

2. I hope there should be universal construction with \pi: TM -> M as universal arrow.

3. As tangent bundle is so ubiquitous there should be nice universal property for it.

With regards, N. Danilov.