For your first question, that is somehow the point with KZ-monads: they don't look idempotent (in the definition) but they do behave a little bit like idempotent monad in practice: for a KZ-monad, an object can only have one structure of algebra, but it is not true that $TT X$ is the same as $ TX$ and not every morphism between the underlying objects is a morphisms of algebra. But every morphism between the underlying object extend in a unique way as a lax morphisms of algebra. While for a really idempotent monad, every morphism of the underlying object is a morphism of algebra.

The typical example of KZ-monad is the free co-completion monad (let say under finite colimits to avoid size problems, but they are not very important):

An algebra is a finitely co-complete category, a morphism of algebra a finite co-limit preserving functor. But the the co-completion of a co-complete category (seen as a category) is not the category itself (hence there is not real "idempotency" of the monad").

First this definition in terms of finite ordinal is exactly A.Kock's definition in 'Monads for which structures are adjoint to units' (other link to the paper) coupled to A.Kock's description of the monoidal 2-category $\Delta$ en term of generator and relations given in Generators and relations for $\Delta$ as a monoidal $2$-category.

A way to explain the relation to idempotency directly on the definition, is that you can say that an idempotent monad is exactly a mondad $T$ where the two natural map $T(X) \rightrightarrows TT(X)$ are equal. A 'pseudo-idempotent' monad (on a 2-category) would be a monad where there is an isomorphism between these two $1$-cell $T(X) \rightrightarrows TT(X)$ satisfying some coherence condition. A lax-idempotent monad (or KZ-monad) is when you just have a non-invertible $2$-cell between these two $1$-cell also satisfying some coherence conditions.

Only pseudo-idempotent monad would really be "2-dimensional generalization of idempotent monad" in the sens that they would satisfies that $TT(X)$ is isomorphic to $T(X)$ for all $X$.

For more detail, have a look to A.Kock's paper linked above where he develope the theory of such monad (he calls them KZ-doctrines)

For your first question, that is somehow the point with KZ-monads: they don't look idempotent (in the definition) but they do behave a little bit like idempotent monad in practice: For a KZ-monad,

• an object can only have one structure of algebra.
• It is not true that $TT X$ is the same as $ TX$
• Not every morphism between the underlying objects of two algebras is a morphisms of algebra.
• Every morphism between the underlying objects of two algebras is in a unique way a lax morphisms of algebra.

While for a really idempotent monad, every morphism of the underlying object is a morphism of algebra.

The typical example of KZ-monad is the free co-completion monad (let say under finite colimits to avoid size problems, but they are not very important):

An algebra is a finitely co-complete category, a morphism of algebra a finite co-limit preserving functor. For every functor between co-complete category you have a natural comparison maps between $Colim F(X_i)$ and $F(colim X_i)$ (which makes $F$ into a lax morphism of algebras) but it is not always true that this map is an isomorphism (this is the case when $F$ is a morphism of algebra). Also the co-completion of a co-complete category (seen as a category) is not the category itself (hence there is not real "idempotency" of the monad").

Note, that this definition in terms of finite ordinal is exactly A.Kock's definition in 'Monads for which structures are adjoint to units' (other link to the paper), which is given in terms of the existence of a certain $2$-cell satisfying some identity, coupled to A.Kock's description of the monoidal 2-category $\Delta$ en term of generator and relations given in Generators and relations for $\Delta$ as a monoidal $2$-category.

A way to explain the relation to idempotency directly on the definition, is that you can say that an idempotent monad is exactly a mondad $T$ where the two natural maps $T (\epsilon_X), \epsilon_{TX} : T(X) \rightrightarrows TT(X)$ are equal. A 'pseudo-idempotent' monad (on a 2-category) would be a monad where there is an isomorphism between these two $1$-cell $T(X) \rightrightarrows TT(X)$ satisfying some coherence condition. A lax-idempotent monad (or KZ-monad) is when you just have a non-invertible $2$-cell between these two $1$-cell also satisfying some coherence conditions. (this is the definition in Kock's paper)

Only a pseudo-idempotent monad would really be "2-dimensional generalization of idempotent monad" in the sens that they would satisfies that $TT(X)$ is isomorphic to $T(X)$ for all $X$.

For more detail, have a look to A.Kock's paper linked above where he develop the theory of such monad (he calls them KZ-doctrines)