It pertains to the classical literature on the subject (see e.g. here) 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)?

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