If $\nabla_{Y}$ defines an integral affine structure on $Y$ (i.e. the holonomy of $\nabla_{Y}$ belongs to $GL(n,{{\bf Z}})$), then $\nabla_{Y}^{\vee}$ defines an integral affine structure on $Y^{\vee}$. As the dual covariant lattice one takes the lattice $(T^{{\bf Z}}_{Y})^{\vee}$, which is dual to $T_{Y}^{{\bf Z}}$ with respect to the metric $g_{Y}$.