# Is my understanding about the smooth structure on tangent bundle accurate?

I'm not sure if I understood the smooth structure on tangent bundle of a smooth manifold, so my question is whether my understanding is correct, or not, so let me explain what I've understood so far.

Let $$TM = \cup_{p\in M} \{p\}\times T_p M,$$ and define $$\sigma_p : T_pM \to \mathbb{R}^n$$ by $$\sigma_p ([c]) = D(\phi\circ c)(0)$$, where $$\phi$$ is a local coordinate chart around $$p$$ and $$c$$ is a smooth path with $$c(0) = p \in M$$.Note that, the definition of the tangent space that I'm using given in the first paragraph of this question.

Let $$U = dom(\phi)$$, and $$\sigma:R \subseteq TM \to \mathbb{R}^n \times \mathbb{R}^n$$ be given by $$\sigma ((p, [c])) = (\phi(p), \sigma_p ([c])) =(\phi(p), D(\phi\circ c)(0)),$$ where $$\pi_1 (R) = U.$$

Now, we define a topology and a smooth structure using $$\sigma$$. One can easily check that, $$\sigma$$ is bijection, since when the first arguments are the same, $$D(\phi\circ c_1)(0) = D(\phi\circ c_2)(0)$$ implies $$[c_1] = [c_2]$$.Hence, we can put a topology on $$TM$$ in a way that $$\sigma$$ that makes $$\sigma$$ continuous by pulling the open sets in the product topology of $$\mathbb{R}^n \times \mathbb{R}^n$$. Since the coordinate charts are compatible, this in fact defines a unique topology on $$TM$$ that is independent of the charts chosen. To show that, observe the following:

Let $$R := \sigma_\phi^{-1} (U \times V)$$ for some open subsets $$U,V\in \mathbb{R}^n$$, i.e $$R$$ be an open subset of $$TM$$ wrt to the topology defined by using $$\phi$$. Then, observe that $$\sigma_\psi (R) = \sigma_\psi \circ \sigma_\phi^{-1} (U\times V) = ( \psi\circ \phi^{-1} (U), D(\psi \circ \phi^{-1})(V)).$$ Since this maps is a diffeomorphism in both coordinates, the image of $$R$$ under $$\phi_\psi$$ is open, hence the topology is independent of the coordinate charts. Moreover, this shows that we can cover $$TM$$ with $$\sigma_\phi$$'s in a compatible manner, hence such maps defines a natural smooth structure on this topological manifold, making into a smooth manifold.

Just to check: in the expression $$\sigma_\psi (R) = \sigma_\psi \circ \sigma_\phi^{-1} (U\times V) = ( \psi\circ \phi^{-1} (U), D(\psi \circ \phi^{-1})(V)),$$ is the second component correct ?