Monads for which Structures are Adjoint to Units

We present here the equational two-dimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that already in 1973 [9], [10] , and the present note is in some sense identical to that, but with some further equational consequences added. The kind of structure introduced in [9], [10] has in the meantime been applied and improved by various authors, notably Street [12] [13], who used the term ”monads with the Kock property” and ”KZ doctrine” (”Kock-Zöberlein”). Some of Street’s improvements are incorporated in our results below. We shall use the term 2-doctrine, for the reason given in Section 2 below. Thus, a 2-doctrine T is an endofunctor T on the 2-category of categories, which is equipped with y : I → T and m : TT → T , just as monads; but the monad laws hold only up to isomorphisms, and these isomorphisms, as well as the further two-dimensional structure, required for the adjointness alluded to in the title, arise out of a single natural transformation