Let $F$ be an association of a smooth vector bundle of rank $n$ to each smooth $n$-dimensional manifold $M$, satisfying: (i) functoriality — smooth maps $f : M \to N$ induce bundle morphisms $f_\#$ with $(f \circ g)_\# = f_\# \circ g_\#$; (ii) for $\mathbb{R}^n$ the bundle is $\mathbb{R}^n \times \mathbb{R}^n$ with $f_\#$ acting as $(f, Df)$; and (iii) restriction to open subsets is compatible. Then $F(M) \cong TM$.