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

In memory of our colleague Pieter Hofstra

For a 2-category K, we consider Street's 2-category Mnd(K) of monads in K, along with Lack and Street's 2-category EM(K) and the identity-on-objects-and-1- cells 2-functor Mnd(K) → EM(K) between them. We show that this 2-functor can be obtained as a "free completion" of the 2-functor 1: K → 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.