Monoid object

Definition

Let $(\mathcal{C}, \otimes, I, \alpha, \lambda, \rho)$ be a monoidal category. A monoid object in $\mathcal{C}$ is an object $M$ together with morphisms $\mu \colon M \otimes M \to M$ (multiplication) and $\eta \colon I \to M$ (unit) such that:

1. associativity: $\mu \circ (\mu \otimes \id_M) = \mu \circ (\id_M \otimes \mu) \circ \alpha_{M,M,M}$ as morphisms $(M \otimes M) \otimes M \to M$;
2. left unit: $\mu \circ (\eta \otimes \id_M) = \lambda_M$ as morphisms $I \otimes M \to M$;
3. right unit: $\mu \circ (\id_M \otimes \eta) = \rho_M$ as morphisms $M \otimes I \to M$.

A morphism of monoid objects $(M, \mu, \eta) \to (M', \mu', \eta')$ is a morphism $f \colon M \to M'$ with $f \circ \mu = \mu' \circ (f \otimes f)$ and $f \circ \eta = \eta'$.