In R.W.R Darling, on differential geometry, an exercise is to construct the cotangent bundle $T^{*}M$ from transition functions $g_{\gamma\alpha}(p)$. A quick analysis suggest using equivalence classes of the form $[p,\alpha,\lambda]$, where
\begin{equation}(p,\alpha,\lambda)\sim(q,\gamma,\mu)\Leftrightarrow p=q, \lambda=(\phi_{\gamma}\circ\phi_{\alpha}^{-1})^{*}\mu\end{equation} where $\phi_{\alpha,\gamma}$ are different overlapping charts on $M$, and $p\in M$, $\lambda\in T^{*}\mathbb{R}^{n}$. Thus a natural choice for the transition functions would be:
\begin{equation}g_{\gamma\alpha}(p)=(\phi_{\gamma}\circ\phi_{\alpha}^{-1})^{*}(\phi_{\alpha}(p))\end{equation}
(This is similar to choosing $d(\phi_{\gamma}\circ\phi_{\alpha}^{-1})$ as transition functions for the tangent bundle.)
How do I prove that these satisfy the last condition for transition functions? Namely,
\begin{equation}g_{\delta\alpha}g_{\alpha\gamma}g_{\gamma\delta}=I\end{equation}