  Given a Hopf algebra H and an algebra A that is an H-module algebra we
consider the category of left H-modules and A-bimodules, where morphisms are
just right A-linear maps (not necessarily H-equivariant). Given a twist F of H
we then quantize (deform) H to H^F, A to A_\star and correspondingly the
category of left H-modules and A-bimodules to the category of left H^F-modules
and A_\star-bimodules. If we consider a quasitriangular Hopf algebra H, a
quasi-commutative algebra A and quasi-commutative A-bimodules, we can further
construct and study tensor products over A of modules and of morphisms, and
their twist quantization.
  This study leads to the definition of arbitrary (i.e., not necessarily
H-equivariant) connections on quasi-commutative A-bimodules, to extend these
connections to tensor product modules and to quantize them to A_\star-bimodule
connections. Their curvatures and those on tensor product modules are also
determined.
