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Consider a pair $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangiansubmanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is acontact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. Weintroduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, withcoefficients in chains of the based loop space of $\Lambda$ and study itsrelation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentationinduced by $L$, $CE^{\ast}(\Lambda)$ can be expressed as the Adams cobarconstruction $\Omega$ applied to a Legendrian coalgebra, $LC_{\ast}(\Lambda)$.We define a twisting cochain:\[\mathfrak{t} \colon LC_{\ast}(\Lambda) \to\mathrm{B} (CF^*(L))^\#\]via holomorphic curve counts, where $\mathrm{B}$denotes the bar construction and $\#$ the graded linear dual. We show undersimply-connectedness assumptions that the corresponding Koszul complex isacyclic which then implies that $CE^*(\Lambda)$ and $CF^{\ast}(L)$ are Koszuldual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between$CE^*(\Lambda)$ and the cobar of the Floer homology of $L$, $\Omega CF_*(L)$.We use the duality result to show that under certain connectivity and locallyfiniteness assumptions, $CE^*(\Lambda)$ is quasi-isomorphic to $C_{-*}(\OmegaL)$ for any Lagrangian filling $L$ of $\Lambda$. Our constructions haveinterpretations in terms of wrapped Floer cohomology after versions ofLagrangian handle attachments. In particular, we outline a proof that$CE^{\ast}(\Lambda)$ is quasi-isomorphic to the wrapped Floer cohomology of afiber disk $C$ in the Weinstein domain obtained by attaching$T^{\ast}(\Lambda\times[0,\infty))$ to $X$ along $\Lambda$ (or, in theterminology of arXiv:1604.02540 the wrapped Floer cohomology of $C$ in $X$ withwrapping stopped by $\Lambda$). Along the way, we give a definition of wrappedFloer cohomology without Hamiltonian perturbations.

Duality between Lagrangian and Legendrian invariants