A monoidal category is a category $\mathrm{𝒞︀}$ together with

1. a functor $\otimes :\mathrm{𝒞︀}\times \mathrm{𝒞︀}\to \mathrm{𝒞︀}$, the tensor product;
2. an object $𝐼$ of $\mathrm{𝒞︀}$, the unit;

3. natural isomorphisms

   $${𝛼}_{𝑥,𝑦,𝑧}:(𝑥\otimes 𝑦)\otimes 𝑧\to 𝑥\otimes (𝑦\otimes 𝑧){𝜆}_{𝑥}:𝐼\otimes 𝑥\to 𝑥{𝜌}_{𝑥}:𝑥\otimes 𝐼\to 𝑥$$

   the associator, left unitor and right unitor;

such that the triangle and the pentagon below commute for all objects $𝑤,𝑥,𝑦,𝑧$.