What is the universal property of the 2-category of monads?

For a 2-category $\mathcal{K}$, we consider Street’s 2-category Mnd($\mathcal{K}$) of monads in $\mathcal{K}$, along with Lack and Street’s 2-category EM($\mathcal{K}$) and the identity-on-objects-and-1-cells 2-functor Mnd($\mathcal{K}$) $\to$ EM($\mathcal{K}$) between them. We show that this 2-functor can be obtained as a “free completion” of the 2-functor $1\text{\hspace{0.17em}⁣}:\mathcal{K}\to \mathcal{K}$. We do this by regarding 2-functors which act as the identity on both objects and 1-cells as categories enriched a cartesian closed category $\mathbf{B}\mathbf{O}$ whose objects are identity-on-objects functors. We also develop some of the theory of $\mathbf{B}\mathbf{O}$-enriched categories.