I am reading about the local system, and a local system $\mathcal{L}$ on $X$ with values in $\mathcal{C}$ is defined as a functor $$ \mathcal{L}:\Pi(X)\to \mathcal{C} $$ where $\Pi(X)$ is the fundamental groupoid of $X$.
In the article, it gives an example of a flat connection: Let $X$ be a smooth manifold and $\mathcal{C}$ be the category of real vector spaces. Let $\mathcal{L}$ be a vector bundle on $X$ equipped with a flat connection. Then it says it is a local system. For a path $p:I\to X$, $\mathcal{L}_p$ should be a parallel transport along $p$. However, I know that in general, the parallel transport along two curves may differ even though when they are in the same homotopy class. So I assumed that there is something with a flat connection. But I am having trouble proving that, when $X$ is a smooth manifold with a flat connection, two curves in a same homotopy class gives same parallel transports. Can you please give me any help?
Thanks in advance.