2-natural transformation?
In general, if $A$ and $B$ are categories (or, more generally, any category-like things, such as objects of some 2-category) equipped with algebraic structure, a lax morphism $f\colon A\to B$ is one which “preserves” the algebraic structure only up to a not-necessarily invertible transformation.
Of course, this transformation goes in one particular direction; a colax morphism is one where the transformation goes in the other direction. The case of 2-monads, below, provides an almost universally applicable way to decide which direction is “lax” and which is “colax”.
Let $T$ be a 2-monad on a 2-category $K$, and let $A$ and $B$ be (strict, pseudo, or ever lax or colax) $T$-algebras. A lax $T$-morphism $f\colon A\to B$ is a morphism in $K$ together with a 2-cell
satisfying some axioms.
If the 2-cell goes in the other direction, then we say $f$ is a colax $T$-morphism (or oplax $T$-morphism). Equivalently, a colax $T$-morphism is a lax $T^{co}$-morphism, where $T^{co}$ is the induced 2-monad on the 2-cell dual $K^{co}$ (see opposite 2-category).
If the 2-cell is invertible, we call $f$ a pseudo or strong $T$-morphism.
Let $W$ be a 2-comonad on $K$, i.e. a 2-monad on the 1-cell dual $K^{op}$, and let $C$ and $D$ be $W$-coalgebras. A lax $W$-morphism $f\colon C\to D$ is a morphism in $K$ together with a 2-cell
satisfying some axioms.
Note that a lax morphism of algebras for the 2-comonad $W$ is a colax morphism of algebras for the 2-monad $W^{op}$. The reason we choose to call this direction for coalgebras “lax” is that if $T$ is a 2-monad with a right adjoint $T^*$, then $T^*$ automatically becomes a 2-comonad such that $T^*$-coalgebras are the same as $T$-algebras, and with the above definition, lax $T$-morphisms coincide with lax $T^*$-morphisms.
A lax monoidal functor is a lax morphism for the 2-monad on Cat whose algebras are monoidal categories. Similarly, an oplax monoidal functor is a colax morphism for this 2-monad.
A lax natural transformation between 2-functors $C\to D$ is a lax morphism for the 2-monad on $[ob(C),D]$ whose algebras are 2-functors (which exists if $D$ is cocomplete and $C$ is small). Similarly, an oplax natural transformation is a colax morphism for this 2-monad. If $D$ is also complete, then this 2-monad has a right adjoint, which then as usual becomes a 2-comonad whose coalgebras are also 2-functors. The above conventions for lax morphisms between coalgebras mean that a lax natural transformation is unambiguously “lax” rather than “colax”, whether we regard the 2-functors as algebras for a 2-monad or coalgebras for a 2-comonad.
Some authors have tried to change the traditional meanings of “lax” and “colax” in this case, but the general framework of 2-monads gives a good argument for keeping it this way (even if in this particular case, oplax transformations are more common or useful).
A lax functor between 2-categories is a lax morphism for the 2-monad on Cat-graphs whose algebras are 2-categories.
A lax algebra for a 2-monad $T$ is a lax morphism $T\to \langle A,A\rangle$ for the 2-monad whose algebras are 2-monads, where $\langle A,A\rangle$ is the codensity monad of the object $A$.
If $T$ is a lax-idempotent 2-monad, then (by definition) every morphism in the underlying 2-category $K$ between (the objects underlying) $T$-algebras has a unique structure of lax $T$-morphism. For instance, every functor between categories with (some class of) colimits is a lax morphism for the 2-monad which assigns those colimits; the unique lax structure map is the canonical comparison $colim (F\circ D) \to F(colim D)$. Such a morphism is strong/pseudo exactly when it preserves the colimits in question.
For any 2-monad $T$, there are a 2-categories:
We have obvious 2-functors
which are bijective on objects, faithful on 1-cells, and locally fully faithful.
Therefore, we can also assemble a number of F-categories of $T$-algebras and any suitable pair of types of $T$-morphism: strict+pseudo, strict+lax, strict+colax, pseudo+lax, or pseudo+colax.
If we want to consider both lax and colax $T$-morphisms together, the natural structure is a double category: there is a straightforward definition of the squares in a double category whose vertical arrows are colax $T$-morphisms and whose horizontal arrows are lax ones. We could then, if we wish, add some “F-ness” to incorporate pseudo and/or strict morphisms as well.
The 2-category $T Alg_p$ is fairly well-behaved; for strict $T$, it admits all strict PIE-limits (if the base 2-category does), and therefore all 2-limits (i.e. bilimits). When $T$ is accessible, $T Alg_p$ admits all 2-colimits as well (but not, in general, many strict 2-colimits).
However, the 2-categories $T Alg_l$ and $T Alg_c$ are not so well-behaved; they do not have many limits or colimits. But once we enhance them to F-categories, they admit all rigged limits. All three 2-categories also admit morphism classifiers; that is, the inclusions $T Alg_s \to T Alg_*$ have left 2-adjoints.