Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$. Let $\mathfrak{g}$ be the Lie algebra of $G$.
Fact: (as can be found in lecture notes here)
Functions $f_{\alpha}:U_{\alpha}\rightarrow \mathfrak{g}$ satisfying
\begin{align*} f_{\alpha}(m)=\text{ad}_{g_{\alpha\beta}(m)}f_{\beta}(m),\quad m\in U_{\alpha}\cap U_{\beta} \end{align*}
define a section $\sigma\in\Gamma(\text{ad}P)$.
They then extend this idea on pg 11 and interpret the above fact as forms $\tau_{\alpha}\in \Omega^1(U_{\alpha};\mathfrak{g})$ satisfying
\begin{align*} \tau_{\alpha}=\text{ad}_{g_{\alpha\beta}}\circ \tau_{\beta} \end{align*}
determine forms in $\Omega^1(U_{\alpha};\text{ad}P)$.
I'm trying to get a better understanding of this extension. I can see it heuristically. The functions $f_{\alpha}$ take a point in $M$ and spit out a vector in $\mathfrak{g}$. Whereas the functions $\tau_{\alpha}$ take in a point in $M$ and a vector in $TM$ and spits out a vector in $\mathfrak{g}$, so it's natural to say this $\tau_{\alpha}$ then defines a form in $\Omega^1(U_{\alpha};\text{ad}P)$.
However, I really want to see a clear argument written down and am struggling to construct one.
Any ideas?
Thanks!