Dear category people, 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.