  We continue our systematic development of noncommutative and nonassociative
differential geometry internal to the representation category of a
quasitriangular quasi-Hopf algebra. We describe derivations, differential
operators, differential calculi and connections using universal categorical
constructions to capture algebraic properties such as Leibniz rules. Our main
result is the construction of morphisms which provide prescriptions for lifting
connections to tensor products and to internal homomorphisms. We describe the
curvatures of connections within our formalism, and also the formulation of
Einstein-Cartan geometry as a putative framework for a nonassociative theory of
gravity.
