A braided monoidal functor is a strong monoidal functor between braided monoidal categories that
preserves their braidings. Write 
for the underlying functor
and 
for the components of its tensor comparison natural
isomorphism. If 
and 
denote the respective braidings, the compatibility condition
is
 |
It must hold for every pair of objects 
and 
of 
(Etingof et al. 2015, p. 196). Both sides are
morphisms from 
to 
.
The strong monoidal functor already includes compatibility with the associativity and unit constraints. Being braided is an additional property of this structure, rather than an additional choice of structure. A braided monoidal equivalence is a braided monoidal functor whose underlying functor has an inverse up to natural isomorphism (Etingof et al. 2015, p. 196).
