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Wondering for the ncatlab I've encountered the following pages: one about lax-morphisms and the other about lax-algebras for $2$-monads.

For what I could get it seems that lax algebras can be characterized as lax morphisms in a category of strict algebras for a strict $2$-monad having as algebras the strict monads.

Could anyone write down more explicitly this construction or link any reference where such characterization is spelled out?

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Ok, after some work I think I've finally get to the end of the construction.

To simplify let's consider just algebras for an endofunctor $T \colon \mathbf C \to \mathbf C$ in the $2$-category $\mathbf C$. An algebra of this sort is a pair $\langle A, m \colon T(A) \to A\rangle$ of on an object $A \in \mathbf C$ and a morphism $m \colon T(A) \to A$.

We know that we can identify object of the $2$-category $\mathbf C$ with functors from the $2$-category $1$ to $\mathbf C$ (where $1$ is the $2$-category having one object, one morphisms and one $2$-morphism). Under this identification the morphisms between objects of $\mathbf C$ correspond one on one to natural transformation between the constant functors (i.e. the elements of $\mathbf {Cat}(1,\mathbf C)$).

So we can equivalently define an algebra for the endo-$2$-functor $F$ as a pair $\langle A \colon 1 \to \mathbf C, m \colon T \circ A \Rightarrow A\rangle$ of a constant functor and a natural transformation.

Now there's a natural bijection in $T$ $$\mathbf{Func}(T \circ A, A) \cong \mathbf {Func}(T,\langle A, A\rangle)$$ where $\langle A,A\rangle$ is the right kan extension of $A$ along itself.

This means that we can identify $T$-algebras over $A$ with natural transformations from the functor $T$ to the functor $\langle A,A \rangle$.

Now if $T$ is a monad a $T$-algebra is an algebra for the endofunctor $T$ which is also compatible with the monad structure (namely the multiplication $\mu \colon T^2 \Rightarrow T$ and the identity $\eta \colon 1_\mathbf{C} \Rightarrow T$) in a lax way.

From this point of view a $T$-algebra become a pair of an object $A \colon 1 \to \mathbf C$ and a natural transformation $m \colon T \circ A \Rightarrow A$ and modifications $\alpha \colon m \circ (1_T * m) \rightarrow m \circ (\mu * 1_A)$ and $\iota \colon 1_A \rightarrow m \circ (\eta * A)$ satisfying some axioms which give coherence conditions for lax-algebras.

All these data live in the $2$-category of ($2$-)functors, ($2$-)natural transformations and modifications.

I haven't done all the computations, still, but I guess that passing all the data to the isomorphism of the right kan extensions we end up with the data that gives a lax-transformation of monads.

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