In Lee's "Introduction to Smooth Manifolds" Problem 11-18:
(a) Let $\operatorname{Vec}_{\mathbb{R}}$ denote the category of real vector spaces and linear maps...
(c) Let $\operatorname{Diff}_1$ be the category of smooth manifolds and diffeomorphisms ...
(d) Let $\mathfrak{X} \colon \operatorname{Diff}_1 → \operatorname{Vec}_{\mathbb{R}}$ be the covariant functor given by $M ↦ \mathfrak{X}(M)$, and let $\mathfrak{X} × \mathfrak{X} \colon \operatorname{Diff}_1 → \operatorname{Vec}_{\mathbb{R}}$ be the covariant functor given by $M ↦ \mathfrak{X}(M) × \mathfrak{X}(M)$, $F ↦ F_* × F_*$ [where $F_*$ is the pushforward, which is well-defined since $F$ is a diffeomorphism]. Show that the Lie bracket is a natural transformation from $\mathfrak{X} × \mathfrak{X}$ to $\mathfrak{X}$.
This answer also draws out the naturality diagram:
$$\require{AMScd}\begin{CD}\mathfrak{X}(M)\times \mathfrak{X}(M) @>F_*\times F_*>> \mathfrak{X}(N)\times \mathfrak{X}(N)\\@VV[\cdot,\cdot]V @VV[\cdot,\cdot]V\\\mathfrak{X}(M) @>F_*>> \mathfrak{X}(N)\end{CD}$$
However, isn't there an error in the problem statement/diagram in that $[\cdot,\cdot]$ isn't an arrow in $\operatorname{Vec}_{\mathbb{R}}$ since it's bilinear and not linear?
The diagram is valid if we implicitly apply the forgetful functor $\operatorname{Vec}_{\mathbb{R}} → \operatorname{Set}$, but a better option to me would be to define the functor $\mathfrak{X} ⊗ \mathfrak{X} \colon \operatorname{Diff}_1 → \operatorname{Vec}_{\mathbb{R}}$ which sends $M ↦ \mathfrak{X}(M) ⊗ \mathfrak{X}(M)$ and $F ↦ F_* ⊗ F_*$, and then for the problem to show that the Lie bracket is a natural transformation from $\mathfrak{X} ⊗ \mathfrak{X}$ to $\mathfrak{X}$, in the sense that the following diagram commutes:
$$\require{AMScd}\begin{CD}\mathfrak{X}(M)⊗ \mathfrak{X}(M) @>F_*⊗ F_*>> \mathfrak{X}(N)⊗ \mathfrak{X}(N)\\@VV[\cdot ⊗\cdot]V @VV[\cdot ⊗ \cdot]V\\\mathfrak{X}(M) @>F_*>> \mathfrak{X}(N)\end{CD}$$
where $[\cdot ⊗\cdot]$ is the unique linear function corresponding to the bilinear function $[\cdot, \cdot]$.
Does that sound right?
