Let $X$ be the open disk of radius $1$ and $U$ the punctured open disk of radius $1$ at the origin. If $u\in U$ and $\pi_1(U,u)=\mathbb{Z}$ is the fundamental group, then we know that categorically, locally constant sheaves on $U$ are the same as $\pi_1(U,u)$-modules through the functor $\mathcal{F}\to\mathcal{F}_u.$
Now, let $F$ be any $\mathbb{Z}$-module, $\mathcal{F}$ the corresponding sheaf, and $j:U\to X$ the embedding. Why is it true that $$ (j_\star\mathcal{F})_{(0,0)}=F^{\mathbb{Z}}, $$ where $j_\star$ is the direct image functor.
I tried to compute this using the definition of $j_\star$ and the direct-limit definition of the stalk, but I have no idea how to compute $\mathcal{F}(V\setminus\{(0,0)\})$ where $V$ is a neighborhood of $(0,0).$ As far as I can tell, even if $V$ is small enough, $\mathcal{F}|_{V\setminus\{(0,0)\}}$ isn't a constant sheaf if $F^\mathbb{Z}\neq F.$
An answer or a reference would be great. I am trying to learn this algebraic topology example so I can apply the same reasoning/intuition to the étale case for the punctured line.
Thanks.