Monoidal category

Definition

A monoidal category is a category $\mathcal{C}$ together with the following data:

1. a functor $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ on the product category, called the tensor product;
2. an object $I$, called the unit object;
3. a natural isomorphism $\alpha$ with components $\alpha_{A,B,C} \colon (A \otimes B) \otimes C \to A \otimes (B \otimes C)$, called the associator;
4. natural isomorphisms $\lambda$ and $\rho$ with components $\lambda_A \colon I \otimes A \to A$ and $\rho_A \colon A \otimes I \to A$, called the left and right unitors.

Here natural means natural as a natural transformation in each object variable. These data must satisfy two axioms for all objects $A, B, C, D$:

1. pentagon: $(\id_A \otimes \alpha_{B,C,D}) \circ \alpha_{A, B \otimes C, D} \circ (\alpha_{A,B,C} \otimes \id_D) = \alpha_{A,B,C \otimes D} \circ \alpha_{A \otimes B, C, D}$ as morphisms $((A \otimes B) \otimes C) \otimes D \to A \otimes (B \otimes (C \otimes D))$;
2. triangle: $(\id_A \otimes \lambda_B) \circ \alpha_{A,I,B} = \rho_A \otimes \id_B$ as morphisms $(A \otimes I) \otimes B \to A \otimes B$.