Monoidal categories

Definition: Monoidal category

A monoidal category $(\mathcal{M}, \otimes, I_\mathcal{M}, a, l, r)$ is a category $\mathcal{M}$ equipped with:

A functor $\otimes : \mathcal{M} \times\ \mathcal{M} \rightarrow \mathcal{M}$ from the product category of $\mathcal{M}$ with itself, called the tensor product.
An object $I_\mathcal{M} \in Ob(\mathcal{M})$ called the unit object or tensor unit.
A natural isomorphism $a : ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))$ with components of the form $a_{X,Y,Z} : (X \otimes Y) \otimes Z \overset{\simeq}{\longrightarrow} X \otimes (Y \otimes Z)$ called the associator.
A natural isomorphism $l : I_\mathcal{M} \otimes (-) \overset{\simeq}{\longrightarrow} (-)$ with components of the form $l_X : I_\mathcal{M} \otimes X \overset{\simeq}{\longrightarrow} X$ called left unitor.
A natural isomorphism $r : (-) \otimes (I_\mathcal{M}) \overset{\simeq}{\longrightarrow} (-)$ with components of the form $r_X : X \otimes I_\mathcal{M} \overset{\simeq}{\longrightarrow} X$ called right unitor.

such that the following two diagrams commute for all objects: