Davilov writes:
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?
From his theorems, it results in particular that Ehresmann's first order velocity functors Tn which associate to a manifold M the vector bundle of
Several authors have been interested in this problem many years ago, that has led to the study of the functors from Man to the category of vector bundles. Such functors are completely characterized in Epstein, "Natural vector bundles", Lecture Notes in Math. 99, Springer 1969, p. 171-195. the 1-jets from R^n to M (in particular T for n=1) are the only such product preserving functors. This result is generalized in an abstract setting to characterize connections in: Bowshell, "Abstract velocity functors", Cahiers de Topologie et Geom. Diff. XII-1 (1971), 57-82. Later on, related (or more general) problems are considered in: Palais & Terng, "Natural bundles have finte order", Topology 16 (1977), 271-277, Epstein & Thurston, "Transformation groups and natural bundles", Proc. London Math. Soc. 38 (1979), 219-236. Eck, "Product preserving functors on smooth manifolds, J. Pure & App. Algebra 42 (1986), 133-140. Kolar, An abstract characterization of the jet spaces, Cahiers de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125. Dupovec & Kolar, On the jets of fibered manifold morphisms, Cahiers de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125. Hoping these old references can be of some interest Best regards Andree C. Ehresmann