promonoidal functor

Definition. Let $\mathbb{V}$ and $\mathbb{W}$ be promonoidal categories. A promonoidal functor $F﹕\mathbb{V}\to \mathbb{W}$ is a functor between the two categories together with transformations $\mu ﹕\mathbb{V}(A;B\otimes C)\to \mathbb{W}(FA;FB\otimes FC),$ $\epsilon ﹕\mathbb{V}(A)\to \mathbb{W}(FA),$ that behave nicely with associators and unitors, satisfying $\lambda ⨾F_{map}=(\mu \times \epsilon )⨾\lambda$, $\rho ⨾F_{map}=(\mu \times \epsilon )⨾\rho$, and $\alpha ⨾(\mu \times \mu )⨾i=(\mu \times \mu )⨾i⨾\alpha$.

Proposition. A promonoidal functor extends to a multifunctor between the induced multicategories.

Tags: multicategory - promonoidal category, multicategory, Promonoidals as multicategories that factor.