We show that, if one allows for curved deformations, the canonical map introduced by Keller and Lowen [Int. Math. Res. Not. IMRN 17 (2009), 3221–3235] between Morita deformations and second Hochschild cohomology of a dg algebra becomes a bijection. We also show that a bimodule induces an equivalence of curved deformations precisely when it induces an equivalence between the respective 1 1 -derived categories. These results, together with the results of Lehmann and Lowen [Filtered derived categories of curved deformations, https://arxiv.org/abs/2402.08660, 2024], offer a solution to the curvature problem for first order deformations.