It is well known that the identity and dual space functors on the category $\mathrm{Vec}_\mathbb{R}$ are not naturally isomorphic because one is covariant and the other contravariant. We can then consider whether the two are dinaturally isomorphic, that is, whether there exists a family $\{\xi_V : V \to V^*\}_{V \in \mathrm{Obj}(\mathrm{Vec}_\mathbb{R})}$ such that for every linear map of real vector spaces $T: V \to W$ the following diagram commutes: $$ \require{AMScd} \begin{CD} V @>{\xi_V}>>V^*\\ @V{T}VV @A{T^*}AA\\ W @>{\xi_W}>>W^* \end{CD} $$ If this holds, by setting $V=W$ and $Tv = 2v$ one can show that $\xi_V$ must be $0$ for all $V$ which is a contradiction.
Does the same result hold for the tangent and cotangent functors for smooth manifolds? I know the easy thing to do is to note that the tangent bundle of $\mathbb{R}^n$ with its canonical smooth structure is both a vector space as well as a smooth manifold, and hence the above proof can be extended to this case as well. But can we show the same thing on the full subcategory of smooth manifolds that are not diffeomorphic to $\mathbb{R}^n$?