In a monoidal category, we can take the tensor product of objects, X ⊗ Y and of morphisms f ⊗ g. Tensor product does not need to be strictly associative on objects, but there is a specified associator, α_ X Y Z : (X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z). There is a tensor unit 𝟙_ C, with specified left and right unitor isomorphisms λ_ X : 𝟙_ C ⊗ X ≅ X and ρ_ X : X ⊗ 𝟙_ C ≅ X. These associators and unitors satisfy the pentagon and triangle equations.

The tensor product of two isomorphisms is an isomorphism.

The tensor product expressed as a functor.

The left-associated triple tensor product as a functor.

The right-associated triple tensor product as a functor.

The functor λ X, 𝟙_ C ⊗ X.

The functor λ X, X ⊗ 𝟙_ C.

The associator as a natural isomorphism.

The left unitor as a natural isomorphism.

The right unitor as a natural isomorphism.