Suppose $V$ is a Banach space and let $j_0\in V\to V^{**}$, $j_1\in V^*\to V^{***}$ be the canonical isometries into the double-dual. If $V$ is reflexive, then $j_0$ is a surjection; moreover one can compute $j_1=(j_0^{-1})^*$ by unraveling the definitions.
But what if $V$ is not reflexive? When does there exist a map $T\in V^{**}\to V$ such that $j_1=T^*$? Since $j_1$ is always injective, $T$ must be a surjection.
Such a $T$ exists iff $\text{im}(j_0)$ is complemented in $V^{**}$.
If the image of $j_0$ is complemented (like, for example, James' space), then $T$ can be the projection from $V^{**}$ onto $\text{im}{(j_0)}$.
Conversely, if $T$ exists, then $j_0T$ must project $V^{**}\to\text{im}{(j_0)}$. But then $V^{**}=\ker{(T)}\oplus\text{im}{(j_0)}$.