Ideal in a Monoid Object

An two-sided ideal of a Monoid Object ( M , η , μ ) 𝑀 𝜂 𝜇 (M,\eta,\mu) in a Monoidal Category ( 𝒱 , ⊗ , I ) 𝒱 tensor-product 𝐼 (\mathcal{V},\otimes,I) is a Subobject a : A ↣ M : 𝑎 ↣ 𝐴 𝑀 a:A\rightarrowtail M such that we have maps l : A ⊗ M → A : 𝑙 → tensor-product 𝐴 𝑀 𝐴 l:A\otimes M\to A , r : M ⊗ A → A : 𝑟 → tensor-product 𝑀 𝐴 𝐴 r:M\otimes A\to A that make the following diagrams commute

M⊗Atensor-product𝑀𝐴{{M\otimes A}}M⊗Mtensor-product𝑀𝑀{{M\otimes M}}A𝐴{A}M𝑀{M}mathrmidifx..⋄lsei⊗a\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes a}l𝑙\scriptstyle{l}μ𝜇\scriptstyle{\mu}a𝑎\scriptstyle{a}

A⊗Mtensor-product𝐴𝑀{{A\otimes M}}M⊗Mtensor-product𝑀𝑀{{M\otimes M}}A𝐴{A}M𝑀{M}a⊗mathrmidifx..⋄lsei\scriptstyle{a\otimes mathrm{id}{ifx..lse\par i}}r𝑟\scriptstyle{r}μ𝜇\scriptstyle{\mu}a𝑎\scriptstyle{a}

If only l 𝑙 l or r 𝑟 r exists, then A 𝐴 A is a left or a right ideal.