It pertains to the classical literature on the subject (see e.g. [here](https://golem.ph.utexas.edu/category/2014/04/on_two-dimensional_monad_theory.html)) the fact that an accessible 2-monad $T$ on a 2-category $\cal K$ induces an adjunction
$$
\mathbf{Strict}\text{-}T\text{-}\textbf{Alg}\leftrightarrows \mathbf{Pseudo}\text{-}T\text{-}\textbf{Alg}
$$
between strict $T$-algebras and pseudo $T$-algebras. This means that for every morphism $(A,a)\to (B,b)$ of $T$-algebras where the relevant diagrams commute up to invertible 2-cells, there is a new $T$-algebra $(A^\text{s}, a^\text{s})$ with a *strict* morphism $(A^\text{s}, a^\text{s}) \to (B,b)$.



> 
> What does this construction become when $T : {\bf Cat}\to {\bf Cat}$ is the 2-monad that sends a small category $A$ into the category $A^\textbf{2}$ of its arrows (for sure a finitary monad on a locally finitely presentable category)?
> 
> 
> 



Strictification of the squaring monad
A strict algebra of the squaring monad is precisely a [strict factorization system](https://ncatlab.org/nlab/show/strict+factorization+system). A normal pseudo-algebra (unit holds strictly) is precisely an orthogonal factorization system. See [here](https://doi.org/10.1016/0022-4049(93)90171-O). The pseudo-algebras will differ by first replacing a morphism by an isomorphic morphism before factoring it.


So that's what the pseudo-algebras look like. You can read off from this a description of the strictified monad.


