Bimodule between Monoid Objects

Let ( 𝒱 , ⊗ , I ) 𝒱 tensor-product 𝐼 (\mathcal{V},\otimes,I) be a Monoidal Category, and ( A , η A , μ A ) 𝐴 subscript 𝜂 𝐴 subscript 𝜇 𝐴 (A,\eta_{A},\mu_{A}) and ( B , η B , μ B ) 𝐵 subscript 𝜂 𝐵 subscript 𝜇 𝐵 (B,\eta_{B},\mu_{B}) be a pair of Monoid Objects. A ( A , B ) 𝐴 𝐵 (A,B) bimodule in 𝒱 𝒱 \mathcal{V} is an object M 𝑀 M equipped with

Such that

A⊗A⊗Mtensor-product𝐴𝐴𝑀{{A\otimes A\otimes M}}A⊗Mtensor-product𝐴𝑀{{A\otimes M}}A⊗Mtensor-product𝐴𝑀{{A\otimes M}}M𝑀{{M}}mathrmidifx..⋄lsei⊗l\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes l}μA⊗mathrmidifx..⋄lsei\scriptstyle{\mu_{A}\otimes mathrm{id}{ifx..lse\par i}}l𝑙\scriptstyle{l}l𝑙\scriptstyle{l}

I⊗Mtensor-product𝐼𝑀{{I\otimes M}}A⊗Mtensor-product𝐴𝑀{{A\otimes M}}M𝑀{M}ηA⊗mathrmidifx..⋄lsei\scriptstyle{\eta_{A}\otimes mathrm{id}{ifx..lse\par i}}λ𝜆\scriptstyle{\lambda}l𝑙\scriptstyle{l}

M⊗B⊗Btensor-product𝑀𝐵𝐵{{M\otimes B\otimes B}}M⊗Btensor-product𝑀𝐵{{M\otimes B}}M⊗Btensor-product𝑀𝐵{{M\otimes B}}M𝑀{{M}}r⊗mathrmidifx..⋄lsei\scriptstyle{r\otimes mathrm{id}{ifx..lse\par i}}mathrmidifx..⋄lsei⊗μB\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes\mu_{B}}r𝑟\scriptstyle{r}r𝑟\scriptstyle{r}

M⊗Itensor-product𝑀𝐼{{M\otimes I}}M⊗Atensor-product𝑀𝐴{{M\otimes A}}M𝑀{M}mathrmidifx..⋄lsei⊗ηB\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes\eta_{B}}ρ𝜌\scriptstyle{\rho}r𝑟\scriptstyle{r}

A⊗M⊗Btensor-product𝐴𝑀𝐵{{A\otimes M\otimes B}}A⊗Mtensor-product𝐴𝑀{{A\otimes M}}M⊗Btensor-product𝑀𝐵{{M\otimes B}}M𝑀{{M}}mathrmidifx..⋄lsei⊗r\scriptstyle{mathrm{id}{ifx..lse\par i}\otimes r}l⊗mathrmidifx..⋄lsei\scriptstyle{l\otimes mathrm{id}{ifx..lse\par i}}r𝑟\scriptstyle{r}l𝑙\scriptstyle{l}

Examples