A \(T\)-monoid in a virtual double category, for a monad \((T, \eta , \mu )\) on a virtual double category, is a monoid in the loose (horizontal) Kleisli virtual double category of \(T\).
It comprises in other words,
- An object \(X_0\),
- a loose morphism \(X : X_0 \nrightarrow TX_0\),
- a unit cell,
- a multiplication cell,
- satisfying unitality and associativity equations.