2-natural transformation?
A lax-idempotent 2-monad encodes a certain kind of property-like structure that a category, or more generally an object of a 2-category, can carry.
The archetypal examples are given by 2-monads $T$ on Cat that take a category $C$ to the free cocompletion $T C$ of $C$ under a given class of colimits – then an algebra $T C \to C$ is a category $C$ with all such colimits, which are of course essentially unique. Moreover, given thus-cocomplete categories $C$ and $D$, a functor $F \colon C \to D$, and a diagram $S$ in $C$, there is a unique arrow $colim T F S \to F(colim S)$ given by the universal property of the colimit. It is this property that lax-idempotence generalizes.
A 2-monad $T$ on a 2-category $K$ is called lax-idempotent if given any two (strict) $T$-algebras $a \colon T A \to A$, $b \colon T B \to B$ and a morphism $f \colon A \to B$, there exists a unique 2-cell $\bar f \colon b \circ T f \Rightarrow f \circ a$ making $(f,\bar f)$ a lax morphism of $T$-algebras:
Dually, a 2-monad $T$ is called colax-idempotent if $f \colon A \to B$ gives rise to a colax $T$-morphism $(f,\tilde f)$:
Lax-idempotent monads are also called Kock–Zöberlein or KZ monads.
A 2-monad $T$ as above is lax-idempotent if and only if for any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\theta_a \colon 1 \Rightarrow \eta A \circ a$ such that $(\theta_a ,1_{1_A})$ are the unit and counit of an adjunction $a \dashv \eta_A$.
(Adapted from Kelly–Lack). The multiplication $\mu_A \colon T^2 A \to T A$ is a $T$-algebra on $T A$, and $\eta_A \colon A \to T A$ is a morphism from the underlying object of $a$ to that of $\mu_A$. So there is a unique $\bar\eta_A \colon \mu_A \circ T \eta_A = 1_{T A} \Rightarrow \eta_A \circ a$ making $\eta_A$ into a lax $T$-morphism. Set $\theta_a = \bar\eta_A$. The triangle equalities then require that:
$a \bar\eta_A \colon a \Rightarrow a \circ \eta_A \circ a = a$ is equal to $1_a$. The composite $a \circ \bar\eta_A$ makes $a \circ \eta_A$ a lax $T$-morphism from $a$ to $a$ (paste $\bar\eta_A$ with the identity square $a \circ \mu_A = a \circ T a$). But $a \circ \eta_A = 1_A$, and $1_a$ also makes this into a lax $T$-morphism, so by uniqueness $a \bar\eta_A = 1_a$.
$\bar\eta_A \eta_A \colon \eta_A \Rightarrow \eta_A \circ a \circ \eta_A = \eta_A$ is equal to $1_{\eta_A}$. But this follows directly from the unit coherence condition for the lax $T$-morphism $\bar\eta_A$.
Conversely, suppose $\theta_a$, algebras $a,b$ on $A,B$ and $f \colon A \to B$ are given. Take $\bar f$ to be the mate of $1_f \colon b \circ T f \circ \eta A = f \Rightarrow f$ with respect to the adjunctions $a \dashv \eta_A$ and $1 \dashv 1$, which is given in this case by pasting with $\theta_a$, so we have that $\bar f = b \circ T f \circ \theta_a$. The mate of $\bar f$ in turn is given by $\bar f \circ \eta_A$, which because mates correspond bijectively is equal to $1_f$. So $\bar f$ satisfies the unit condition.
Consider the diagrams expressing the multiplication condition: because $a \circ \mu_A = a \circ T a$ (and the same for $b$), their boundaries are equal, so we have 2-cells $\alpha, \beta \colon b \circ T b \circ T^2 f \Rightarrow f \circ a \circ T a$. Their mates under the adjunction $(T\theta_a, 1) \colon T a \dashv T\eta_A$ are given by pasting with $T \eta_A$. One is $\bar f$ pasted with $T \bar f \circ T \eta_A = T(f \circ \eta_A) = T 1_f = 1_{T f}$, and the other is given by composing $T \eta_A$ with the identity $\mu_B \circ T^2 f = T f \circ \mu_A$ (and then pasting with $\bar f$), but because $\mu_A \circ T \eta_A = 1_{T A}$ this is also equal to $1_{T f}$. The two original 2-cells are hence equal, because their mates are equal, and so $\bar f$ is indeed a lax $T$-morphism.
Since $T$‘s multiplication $\mu$ makes $T$ itself into a (generalized) $T$-algebra, the above implies (and in fact is implied by) the requirement that there exist a modification $\ell \colon 1_{T^2} \to \eta T \circ \mu$ making $(\ell,1) \colon \mu \dashv \eta T$. Conversely, given an algebra $a \colon T A \to A$, the 2-cell $\theta_a$ is given by $T a \circ \ell_A \circ T \eta_A$.
A different but equivalent condition is that there be a modification $d \colon T \eta \to \eta T$ such that $d \eta = 1$ and $\mu d = 1$; and given $\ell$ as above, $d$ is given by $\ell \circ T \eta$.
These various conditions can also be regarded as ways to say that the Eilenberg-Moore adjunction for $T$ is a lax-idempotent 2-adjunction. Thus, $T$ is a lax-idempotent 2-monad exactly when this 2-adjunction is lax-idempotent, and therefore also just when it is the 2-monad induced by some lax-idempotent 2-adjunction.
Dually, for $T$ to be colax-idempotent, it is necessary and sufficient that any of the following hold.
For any $T$-algebra $a \colon T A \to A$ there is a 2-cell $\zeta_a \colon \eta_A \circ a \Rightarrow 1$ such that $(1,\zeta_a) \colon \eta_A \dashv a$.
There is a modification $m \colon \mu \circ \eta T \to 1$ making $(1,m) \colon \eta T \dashv \mu$.
There is a modification $e \colon \eta T \to T\eta$ such that $e\eta = 1$ and $\mu e = 1$.
Theorem 1 gives a necessary condition for an object $A$ to admit a $T$-algebra structure, namely that $\eta_A : A \to T A$ admit a left adjoint with identity counit. In the case of pseudo algebras, this necessary condition is also sufficient.
To give a pseudo $T$-algebra structure on an object $A$ is equivalently to give a left adjoint to $\eta_A : A\to T A$ with invertible counit.
In particular, an object admits at most one pseudo $T$-algebra structure, up to unique isomorphism. Thus, $T$-algebra structure is property-like structure.
In many cases it is interesting to consider the pseudo $T$-algebras for which the algebra structure $T A \to A$ has a further left adjoint, forming an adjoint triple. Algebras of this sort are sometimes called continuous algebras.
As mentioned above, the standard examples of lax-idempotent 2-monads are those on $Cat$ whose algebras are categories with all colimits of a specified class. In this case, the 2-monad is a free cocompletion operation. Dually, there are colax-idempotent 2-monads which adjoin limits of a specified class. A converse is given by (PowerCattaniWinskel), who show that a 2-monad is a monad for free cocompletions if and only if it is lax-idempotent and the unit $\eta$ is dense (plus a coherence condition).
Another important example of a colax-idempotent monad is the monad on $Cat/B$ that takes $p \colon E \to B$ to the projection $B/p \to p$ out of the comma category. The algebras for this monad are Grothendieck fibrations over $B$; see also fibration in a 2-category. The monad $p \mapsto p/B$ is lax-idempotent, and its algebras are opfibrations.
This latter is actually a special case of a general situation. If $T$ is a (2-)monad relative to which one can define generalized multicategories, then often it induces a lax-idempotent 2-monad $\tilde{T}$ on the 2-category of such generalized multicategories (aka “virtual $T$-algebras”), such that (pseudo) $\tilde{T}$-algebras are equivalent to (pseudo) $T$-algebras. When $T$ is the 2-monad whose algebras are strict 2-functors $B\to Cat$ and whose pseudo algebras are pseudofunctors $B\to Cat$, then a virtual $T$-algebra is a category over $B$, and it is a pseudo $\tilde{T}$-algebra just when it is an opfibration. Similarly, there is a lax-idempotent 2-monad on the 2-category of multicategories whose pseudo algebras are monoidal categories, and so on.
Max Kelly, Steve Lack, On property-like structures, TAC 3(9), 1997.
Andres Kock, Monads for which structures are adjoint to units, JPAA 104:41–59, 1995.
A. J. Power, G. L. Cattani, G. Winskel, A representation result for free cocompletions, JPAA 151:273–286, 2000 doi
Marmolejo–Wood, Kan extensions and lax idempotent pseudomonads, TAC 26, p. 1–29 (2011)