Let $(M,\mathcal{F})$ be a foliated manifold. Take a leaf $L$, two points $x,y\in L$ and a path $\gamma$ from x to y.
I consider the case where $\gamma([0,1])\subset U$, $U$ is the domain of some foliated chart. The holonomy of $\gamma$ with respect to some transversal sections T and S containing x,y respectively, is the germ of a (locally defined) diffeomorphism $f:(T,x)\rightarrow (S,y)$, with the property that for any $x'$ in the domain of $f$ we have that $x' and f(x')$ are contained in the same plaque of $U$.
I believe that the above property is what makes the above definition of the holonomy of $\gamma$ intependent of the choice of $U$. My problem is that i cannot write the details down explicitly. I would be grateful for any ideas or advice, as I wish to increase my intuiton on the notion of holonomy.