Fixing an orientation on $L$, then $CF^{0}(L,L^{\prime})$ is naturally dual to $CF^{n}(L^{\prime},L)$. The local system factor $\Hom(E,E^{\prime})$ is naturally dual to $\Hom(E^{\prime},E)$. Given a Lagrangian path $\Lambda_{p}$ associated to a Lagrangian intersection $p$, the reverse path is also associated with a determinant line bundle, which can be identified with

$$o_{p}^{\vee}\otimes\lambda(TL),$$

since the two half planes with Lagrangian boundaries can be glued to a disk, such that the det line of the Cauchy-Riemann operator is canonically isomorphic to $\lambda(TL)$.