may also be construed as follows. In addition to what we have above, we should consider the adjunction freely generated by a different object on the $\cal  A$ side, which altogether gives us four disjoint categories. These four categories are isomorphic respectively to the categories *Hom*$({\cal A},{\cal A})$, *Hom*$({\cal A},{\cal B})$, *Hom*$({\cal B},{\cal B})$ and *Hom*$({\cal B},{\cal
A})$ that may be found in the 2-categorial approach of [@A74] and the other references above. Roughly speaking, one has only to understand our freely generated objects as 1-cells, and add 0-cells, to pass to the 2-categorial approach. In contradistinction to that approach, we restrict ourselves to syntactically constructed free adjunctions within the category *Cat*, and we make explicit the free generators, but the mathematical content is essentially the same. (The mathematical content changes by moving to a new level of categorification with the pseudoadjunctions of [@V92] and [@L00].)

We give a new simple proof of the following result of [@A74] (*Corollaire* 2.8), which connects the category $\cal  M$ of the free monad defined at the end of the preceding section with the category $\cal  B$ of the adjunction freely generated by $0$ on the $\cal  B$ side. This result is interesting for 