Dear Steve, Thank you for your precious informations (I have never read the article "two-dimensional ..."). But my interest is really in the case of non necessarily 2-monad over Cat and I am convinced that the adjonction j -| i exists, with or without coherences (for the pseudomorphisms). Best regards, Albert Steve Lack <s.lack@uws.edu.au> a écrit :

Dear Albert,

In the case where T is a 2-monad with rank, and the pseudomorphisms are (as usual) assumed to be coherent, then this was proved by Blackwell-Kelly-Power in the paper 2-dimensional monad theory. (Here "rank" means that the 2-functor T preserves alpha-filtered colimit for some alpha - without some such assumption, I don't know how you can prove that the left adjoint j exists, and I suspect it does not.)

If T has rank but pseudomorphisms are not required to be coherent, then the adjoint j will exist, but the morphism A-->ijA need not be an equivalence (take the identity monad for example).

If T is not even a 2-monad then I'm not sure what coherence of the 2-cells would mean, but in any case there will be problems.

Regards,

Steve Lack.

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