The composition of morphisms in a monoidal category corresponds to an associative binary operation

For a monoidal category $M$ each morphism in $M(m, m)$ corresponds to the operation defined for the associated monoidal set on one of the elements.

For example, in the set of natural numbers $\mathbb{N}$ there will be a morphism corresponding to each partially-applied operation ${+0, +1, ..., +n}$.

Composing any two of these morphisms corresponds to performing the operation - the operation is associative because composition of morphisms is associative.

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• A monoid has an associative binary operation
  • The composition of morphisms in a monoidal category corresponds to an associative binary operation.
• The identity morphism in a monoidal category corresponds to the unit element
  • Because the composition of morphisms in a monoidal category corresponds to an associative binary operation and composition with the identity morphism conforms to the law of identity, the unit element being used as an operand in the associative binary operation is the same thing as composition with the identity morphism.
• A set-theory monoid and a category-theory monoid are the same thing
  • The composition of morphisms in a monoidal category corresponds to an associative binary operation.