Let $\mathcal{A}$ be a locally constant sheaf on a topological space $X$ and let $\sigma:\Delta_p\to X$ denote a singular $p$-simplex. Writing the pullback of $\mathcal A$ by $\sigma$ as $\sigma^\ast(\mathcal A)$, Bredon's book on sheaf theory (page 26 in the second edition) says:
Since $\mathcal{A}$ is locally constant and $\Delta_p$ is simply connected, $\sigma^{\ast}(\mathcal{A})$ is a constant sheaf on $\Delta_p$.
(Emphasis mine).
I realize this is pretty basic, but I can't seem to figure out why the simply-connectedness of $\Delta_p$ enters into this. Is it in general true that the pullback of a locally constant sheaf by a continuous function whose domain is simply connected is a constant sheaf on said domain? Could someone hint at a proof for this?