  We systematically study noncommutative and nonassociative algebras A and
their bimodules as algebras and bimodules internal to the representation
category of a quasitriangular quasi-Hopf algebra. We enlarge the morphisms of
the monoidal category of A-bimodules by internal homomorphisms, and describe
explicitly their evaluation and composition morphisms. For braided commutative
algebras A the full subcategory of symmetric A-bimodule objects is a braided
closed monoidal category, from which we obtain an internal tensor product
operation on internal homomorphisms. We describe how these structures deform
under cochain twisting of the quasi-Hopf algebra, and apply the formalism to
the example of deformation quantization of equivariant vector bundles over a
smooth manifold. Our constructions set up the basic ingredients for the
systematic development of differential geometry internal to the quasi-Hopf
representation category, which will be tackled in the sequels to this paper,
together with applications to models of noncommutative and nonassociative
gravity such as those anticipated from non-geometric string theory.
