In this question $(\mathcal{V}, \otimes, e)$ is a (bi)complete symmetric monoidal category.

We have an adjunction $$\mathscr{l}: \mathsf{Cat} \leftrightarrows \mathcal{V}\text{-}\mathsf{Cat} :(-)_0,$$ induced by the *change of enrichment* as discussed for example between Ex. 3.2 and 3.3 [here][1]. Let me call $\mathsf{T}_{\mathcal{V}}$ the induced monad on $\mathsf{Cat}$.

> **Q1.** Do we know, or can we characterize, when this adjunction is monadic?

> **Q2.** When it is not monadic, how should I think about an algebra for $\mathsf{T}_{\mathcal{V}}$?


Sometimes one reads that being an additive category (i.e. being $\mathsf{Ab}$-enriched) is more a property than a structure, see for example [this question][2] which actually motivated the present question, but I am sure you met such statements in your life. 


> **Q3.** Does this mean that the monad $\mathsf{T}_{\mathsf{Ab}}$ is lax-idempotent, or it is some different behavior?

> **Q4.** Do you have other examples, or a characterization, or a sufficient condition such that being a $\mathcal{V}$-category is a *property*, in the same way this is true for additive categories? (I am aware this is not a well-posed question). 



  [1]: https://ncatlab.org/nlab/show/change+of+enriching+category
  [2]: https://mathoverflow.net/questions/395399/relation-between-ind-completion-and-additive-ind-completion/395401#395401