  We use a version of Haboush's theorem over complete local Noetherian rings to
prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and
separable (braided, symmetric) fusion categories from characteristic $p$ to
characteristic zero (arXiv/math:0203060, Section 9), showing that, moreover,
any isomorphism between such structures can be reduced modulo $p$. This fills a
gap in arXiv/math:0203060, Subsection 9.3. We also show that lifting of
semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove
that lifting induces an isomorphism on Picard and Brauer-Picard groups.
Finally, we show that a subcategory or quotient category of a separable
multifusion category is separable (resolving an open question from
arXiv/math:0203060, Subsection 9.4), and use this to show that certain classes
of tensor functors between lifts of separable categories to characteristic zero
can be reduced modulo $p$.
