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.
The composition of morphisms in a monoidal category corresponds to an associative binary operation.
Similarly, the identity morphism in a monoidal category corresponds to the unit element