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:

$\otimes :\operatorname {Ob} {\mathcal {C}}\times \operatorname {Ob} {\mathcal {C}}\to \operatorname {Ob} {\mathcal {C}}$

along with, for every $A,B,C,D\in \operatorname {Ob} {\mathcal {C}}$, 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: