# A monoid has a unit element¶

A monoid has a unit element u which interacts with the associative binary operation defined for its set, \otimes, in the following way

\exists u \forall a. u \otimes a = a \otimes u = a

The identity morphism in a monoidal category corresponds to the unit element

## Backlinks¶

- The identity morphism in a monoidal category corresponds to the unit element
- The unit element defined for a set-theory monoid corresponds to the identity morphism within a monoidal category.

- A set-theory monoid and a category-theory monoid are the same thing
- The set theory definition and the category theory definition of monoids are different views over the same concept. Category theory defines a monoid as any category with a single object. Set theory defines a monoid as a set with an associative binary operation and a unit element.