Monoidal category

This article defines a notion of a category along with some additional structure.
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Definition

A monoidal category is a category $\mathcal{C}$ equipped with an additional operation, called a monoidal operation:

$\otimes :\mathrm{O}\mathrm{b}\mathcal{C}\times \mathrm{O}\mathrm{b}\mathcal{C}\to \mathrm{O}\mathrm{b}\mathcal{C}$

along with, for every $A,B,C,D\in \mathrm{O}\mathrm{b}\mathcal{C}$, a map:

$\otimes :\mathcal{C}(A,B)\times \mathcal{C}(C,D)\to \mathcal{C}(A\otimes C,B\otimes D)$

satisfying the following conditions:

• Functorial in both variables:
• Has an identity upto natural isomorphism:
• Associative upto natural isomorphism: