Free dual spaces and free Banach lattices

The relation between the free Banach lattice generated by a Banach space and free dual spaces is clarified. In particular, it is shown that for every Banach space $E$ the free $p$-convex Banach lattice generated by $E^{**}$, denoted $FBL^p[E^{**}]$, admits a canonical isometric lattice embedding into $FBL^p[E]^{**}$ and $FBL^p[E^{**}]$ is lattice finitely representable in $FBL^p[E]$. Moreover, we also show that for $p>1$, $FBL^p[E]^{**}$ can actually be considered as the free dual $p$-convex Banach lattice generated by $E$, whereas for $p=1$ this happens precisely when $E$ does not contain complemented copies of $\ell_1$.