  We prove the equivalence of the deformation theory for a higher dimensional
Calabi--Yau manifold and that for its dg category of perfect complexes by
giving a natural isomorphism of the deformation functors. As a consequence, the
dg category of perfect complexes on a versal deformation of the original
manifold provides a versal Morita deformation of its dg category of perfect
complexes. Besides the classical uniqueness up to \'etale neiborhood of the
base, we prove another sort of uniqueness of versal Morita deformations.
Namely, given a pair of derived-equivalent higher dimensional Calabi--Yau
manifolds, the dg categories of perfect complexes of their algebraic
deformations over a common base, which always exist, become quasi-equivalent
close to effectivizations. Then the base change along the corresponding first
order approximation yields quasi-equivalent versal Morita deformations. We
introduce the generic fiber of the versal Morita deformation as a Drinfeld
quotient, which is quasi-equivalent to the dg category of perfect complexes on
the generic fiber of the versal deformation.
