Verify that for all $u\in\Hom(E)$: $Ju=u^{**}J$. Thus identifying $E$ with $E^{**}$ (via $J$), i.e. we consider $Jx$ and $x$ to be the same, we may simply write: $u^{**}=u$.
For all $x\in E$: $Ju(x)\in E^{**}$ and by definition of $J$ and $u^{*}$ we get for all $x^*\in E^*$: $J(u(x))(x^*)=x^*(u(x))=u^*(x^*)(x)$; on the other hand by definition of $u^{**}$ and $J$: $u^{**}(J(x))(x^*)=J(x)(u^*(x^*))=u^*(x^*)(x)$.