Summary: We give a combinatorial proof of the quasi-invertibility of \(\widehat{CFDD}(\mathbb{I}_{\mathcal{Z}})\) in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra \(\mathcal{A}(\mathcal{Z})\), for each pointed matched circle \(\mathcal{Z}\). We do this by giving an explicit description of a rank 1 model for \(\widehat{CFAA}(\mathbb{I}_{\mathcal{Z}})\), the quasi-inverse of \(\widehat{CFDD}(\mathbb{I}_{\mathcal{Z}})\). To obtain this description we apply homological perturbation theory to a larger, previously known model of \(\widehat{CFAA}(\mathbb{I}_{\mathcal{Z}})\).