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This is a follow up of this question I asked some time ago regarding the tangent space functor. I am wondering though if there is a simpler way that this situation can be characterized without formally invoking all of the machinery of vector bundles.

Given any manifolds $M$ and $N$ and $p \in M$, any smooth map $F$ between $M$ and $N$ induces a linear map, the pushforward, $F_{*}:T_pM \rightarrow T_{F(p)}N$ between tangent spaces such that $F_*(X_p)f = X_p(f \circ F)$ for all tangent vectors $X_p \in T_pM$ and smooth functions $f \in C^{\infty}(N)$. The pushforward satisfies the functorial properties $(G \circ F)_{*} = G_{*} \circ F_{*}$ and $(1_M)_* = 1_{T_pM}$.

As I understand the situation, the only thing needed to turn this setup into a functor is a way of assigning each point $p \in M$ to the tangent space $T_pM$. The tangent space $T_pM$ is an element of the tangent bundle $TM$ considered as a disjoint union of all tangent spaces to $M$. Furthermore, at each point $p \in M$ we can use the natural inclusion to assign $M$ its tangent space $T_pM$. It seems this assignment with the pushforward comprises a functor between smooth manifolds and tangent spaces. Is my understanding of the situation accurate? By construction, I realize that this isn't precisely the same tangent functor that Lee defines in Introduction to Smooth Manifolds but is my assement accurate that it is, in fact, a functor?

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A "tangent space" is just a vector space. In your other post you found out that you have a functor to vector bundles: how does a vector space relate to a vector bundle? With this in mind you should just check if you "functor" preserves composition and unit, then it will be an actual functor. –  Andy Jul 9 '12 at 17:39
What you describe is a functor from the category of pointed manifolds and smooth basepoint-preserving maps to the category of vector spaces and linear maps. Namely, you assign $T_pM$ to a pair $(M,p)$ and $f_*: T_pM \to T_qN$ to a smooth base point-preserving map $f: (M,p) \to (N,q)$. –  Alexander Thumm Jul 9 '12 at 17:44
"It seems this assignment (...) comprises a functor between smooth manifolds and tangent spaces." Please define this 'category of tangent spaces'. –  wildildildlife Jul 9 '12 at 19:43

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