Let $Q$ be a regular local ring, $I \subset Q$ a complete intersection ideal and $R=Q/I$. The Koszul complex $K$ is a dg algebra resolution of $R$ over $Q$. We use Koszul duality in form algebra-coalgebra duality, the Koszul complex is not augmented as dg algebra, hence the dual coalgebra is curved. We show that Eisenbud-Shmash resolution follows immediately from the standard adjunction of functors for this form of Koszul duality. Dual of the Koszul complex is naturally a non-coaugmented dg coalgebra, the Koszul dual is curved polynomial algebra. We show that twisting cochains of this curved Koszul duality appear naturally in the lifting problem for complexes of free modules.