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.

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.

More generally, in any $2$-category $\mathfrak{C}$ we have a forgetful functor $$?:[{\bf 1},\mathfrak{C}]_{lax}\to\mathfrak{C}$$ given by sending a monad on an object to its underlying object.

When does this $2$-functor have a left/right adjoint?

I think one of them is again given by sending an object $C$ to the trivial monad $(1_C,1_{1_C},1_{1_C})$, but I'm curious if the other always/never/sometimes exists (and if it's only sometimes what conditions it's equivalent to).