# Monoid (category theory)

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In category theory, a monoid (or monoid object) ${\displaystyle (M,\mu ,\eta )}$ in a monoidal category ${\displaystyle (\mathbf {C} ,\otimes ,I)}$ is an object M together with two morphisms

such that the pentagon diagram

and the unitor diagram

commute. In the above notations, Template:Serif is the unit element and ${\displaystyle \alpha }$, ${\displaystyle \lambda }$ and ${\displaystyle \rho }$ are respectively the associativity, the left identity and the right identity of the monoidal category C.

Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop.

Suppose that the monoidal category C has a symmetry ${\displaystyle \gamma }$. A monoid ${\displaystyle M}$ in C is symmetric when

${\displaystyle \mu \circ \gamma =\mu }$.

## Categories of monoids

Given two monoids ${\displaystyle (M,\mu ,\eta )}$ and ${\displaystyle (M',\mu ',\eta ')}$ in a monoidal category C, a morphism ${\displaystyle f:M\to M'}$ is a morphism of monoids when

The category of monoids in C and their monoid morphisms is written MonC.