What is the universal property of the 2-category of monads? (Q6593816)

\textit{R. Street} [J. Pure Appl. Algebra 2, 149--168 (1972; Zbl 0241.18003)] introduced, for a given 2-category \(\mathcal{K}\), a 2-category \(\mathrm{Mnd}\left( \mathcal{K}\right) \)\ whose objects are the monads in \(\mathcal{K}\). Some thirty years later, a variant \(\mathrm{EM} \left( \mathcal{K}\right) \)\ of \(\mathrm{Mnd}\left( \mathcal{K}\right) \)\ was introduced in [\textit{S. Lack} and \textit{R. Street}, J. Pure Appl. Algebra 175, No. 1--3, 243--265 (2002; Zbl 1019.18002)], having the same objects and 1-cells, but a different notion of 2-cell. There is a 2-functor\N\[\N\mathrm{Mnd}\left( \mathcal{K}\right) \rightarrow\mathrm{EM}\left( \mathcal{K}\right)\N\]\Nbetween them, which acts as the identity on objects and 1-cells. It is still the case that the composite 2-functor\N\[\N\mathcal{K}\rightarrow\mathrm{Mnd}\left( \mathcal{K}\right) \rightarrow \mathrm{EM}\left( \mathcal{K}\right)\N\]\Nhas a right adjoint just when \(\mathcal{K}\)\ admits Eilenberg-Moore objects, while \(\mathrm{EM}\left( \mathcal{K}\right) \)\ is the free completion of \(\mathcal{K}\)\ under Eilenberg-Moore objects. This paper is concerned with the question whether the original \(\mathrm{Mnd}\left( \mathcal{K}\right) \)\ itself has a universal property.\N\NThe authors note that there is a cartesian closed category \(\boldsymbol{BO} \)\ whose objects are identity-on-objects functors and whose morphisms are commutative squares of functors. A \(\boldsymbol{BO}\)-enriched category is essentially the same as a 2-functor acting as the identity on objects and 1-cells. After developing a little of the theory of \(\boldsymbol{BO}\)-enriched category theory, they show how every \(\boldsymbol{Cat}\)-enriched weight gives rise to a corresponding \(\boldsymbol{BO}\)-enriched weight, so that we have a \(\boldsymbol{BO}\)-enriched notion of \(\boldsymbol{BO}\)-enriched notion of Eilenberg-Moore object. It is finally shown that\N\[\N\mathrm{Mnd}\left( \mathcal{K}\right) \rightarrow\mathrm{EM}\left( \mathcal{K}\right)\N\]\Nis the free completion of\N\[\N1:\mathcal{K}\rightarrow\mathcal{K}\N\]\Nunder these \(\boldsymbol{BO}\)-enriched Eilenberg-Moore objects.