For the purpose of this post, we will identify a monad $T$ on a category $\mathcal{C}$ with a lax $2$-functor $T:{\bf 1}\to\mathfrak{Cat}$ such that $T(*)=\mathcal{C}$.

There is an obvious forgetful functor $$?:[{\bf 1},\mathfrak{Cat}]_{lax}\to\mathfrak{Cat}$$ sending a monad to its underlying category.

Does this $2$-functor have a left or right adjoint?

I think one of the two is given by sending a category $\mathcal{C}$ to the trivial monad $(1_\mathcal{C},1_{1_\mathcal{C}},1_{1_\mathcal{C}})$ on $\mathcal{C}$, but if the other adjoint exists it might be more interesting.