Let formula _ 1 and formula _ 2 be natural categories . A monoidal functor from formula _ 3 to formula _ 4 consists of a functor formula _ 5 together with a monoidal transformation .
Let formula _ 1 and formula _ 2 be monoidal categories . A monoidal functor from formula _ 3 to formula _ 4 consists of a functor formula _ 5 together with a natural transformation