Category of Monoid Objects

The category of monoid objects $\mathrm{Mon}(\mathcal{V})$ in a Monoidal Category $(\mathcal{V},\otimes,I)$ is the category where

• Objects are Monoid Objects
• Morphisms are Homomorphism of Monoid Objects

Properties

• We can extend the category of monoid objects to the Double Category of Bimodules when $\mathcal{V}$ has Coequalisers. Alternatively, if $\mathcal{V}$ lacks the requisite Coequalisers, we can still consider the Virtual Double Category of Bimodules.