question on pseudomorphisms

Dear all, Has the following question been already studied and, if it is the case (it is my opinion), where ? Let T be a monad on Cat (not necessarily a 2-monad), C the category of T-algebras and C' the catégorie with the same objets but with pseudomorphisms (morphisms up to natural isomorphisms --- eventually with coherences). The inclusion i : C --> C' has a left adjoint j : C' --> C. The question is : for all T-algebra A, is the canonical morphisms m : A --> i(j(A)) an equivalence (of the underlying categories) ? Regards, Albert [For admin and other information see: http://www.mta.ca/~cat-dist/ ]