Direct image from an open restriction (${}_U\mathcal{F}=j_{\ast }(\mathcal{F}{|}_U)$)

For an open inclusion $j:U\hookrightarrow X$, this sheaf has sections $\mathcal{F}(V\cap U)$ on an open set $V$. Restriction defines $\mathcal{F}\to {}_U\mathcal{F}$. Its stalks outside $U$ may be nonzero, so it differs from extension by zero.