A construction of dual box

Let $\mathtt{R}$ be a quasi-hereditary algebra, $\mathscr{F}(\Delta)$ and $\mathscr{F}(\nabla)$ its categories of good and cogood modules correspondingly. In [6] these categories were characterized as the categories of representations of some boxes $\mathscr{A}=\mathscr{A}_{\Delta}$ and $\mathscr{A}_{\nabla}$. These last are the box theory counterparts of Ringel duality [8]. We present an implicit construction of the box $\mathscr{B}$ such that $\mathscr{B}-\mathrm{mo}$ is equivalent to $\mathscr{F}(\nabla)$.