# nLab bilax monoidal functor

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## Definition

A bilax monoidal functor is a functor $F : C \to D$ between categories equipped with the structure of braided monoidal categories that is both a lax monoidal functor as well as an oplax monoidal functor with natural transformations

$F(x) \otimes F(y) \stackrel{\overset{\Delta_{x,y}}{\leftarrow}}{\underset{\nabla_{x,y}}{\to}} F(x \otimes y)$

satisfying two compatibility conditions:

• braiding For all $a,b,c,d \in C$ the following diagram commutes

$\array{ && F(a \otimes b) \otimes F(c \otimes d) \\ & \swarrow && \searrow \\ F(a \otimes b \otimes c \otimes d) &&&& F(a) \otimes F(b) \otimes F(c) \otimes F(d) \\ \downarrow &&&& \downarrow \\ F(a \otimes c \otimes b \otimes d) &&&& F(a) \otimes F(c) \otimes F(b) \otimes F(d) \\ & \searrow && \swarrow \\ && F(a \otimes c) \otimes F(b \otimes d) }$
• unitality (…)

Definition 3.3 in