  Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2,..., n\}$ and
$g^L$ be its Langlands dual. It is conjectured that for each $i \in I \setminus
\{0\}$ the affine Lie algebra $g$ has a positive geometric crystal whose
ultra-discretization is isomorphic to the limit of certain coherent family of
perfect crystals for $g^L$. We prove this conjecture for $i=2$ and $g =
A_n^{(1)}$.
