  By slicing the Heegaard diagram for a given $3$-manifold in a particular way,
it is possible to construct $\mathcal{A}_{\infty}$-bimodules, the tensor
product of which retrieves the Heegaard Floer homology of the original
3-manifold. The first step in this is to construct algebras corresponding to
the individual slices. In this paper, we use the graphical calculus for
$\mathcal{A}_{\infty}$-structures introduced in arXiv:2009.05222v3 to construct
Koszul dual $\mathcal{A}_{\infty}$ algebras $\mathcal{A}$ and $\mathcal{B}$ for
a particular star-shaped class of slice. Using $\mathcal{A}_{\infty}$-bimodules
over $\mathcal{A}$ and $\mathcal{B}$, we then verify the Koszul duality
relation.
