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

For a 2-category $\mathcal{K}$, we consider Streets 2-category $\mathbf{Mnd}(\mathcal{K})$ of monads in $\mathcal{K}$, along with Lack and Streets $2$-category $\mathbf{EM}(\mathcal{K})$ and the identity-on-objects-and-$1$-cells $2$-functor $\mathbf{Mnd}(\mathcal{K}) → \mathbf{EM}(\mathcal{K})$ between them. We show that this $2$-functor can be obtained as a free completion of the 2-functor $1:\mathcal{K}→\mathcal{K}$. 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 BO whose objects are identity-on-objects functors. We also develop some of the theory of BO-enriched categories.