Let 
and 
be monoidal categories. A lax monoidal functor
from 
to 
is a functor 
together with a morphism 
and a natural
transformation whose components are
 |
These structure morphisms must be compatible with the associators and unitors of the two categories.
If 
and all the 
are isomorphisms, then 
is a strong monoidal functor, often called simply a monoidal
functor. If the structure morphisms are identity morphisms, then 
is a strict monoidal functor.
