adjoints to lax-idempotent algebra structures

Let T be a lax-idempotent (i.e. Kock-Zoberlein) 2-monad (or pseudomonad). Then to give a pseudo T-algebra structure on an object A is to give a left adjoint a : TA -> A to the unit e : A -> TA. Has anyone studied and/or named the class of T-algebras for which the algebra structure map admits a further left adjoint? In examples, this seems to be a sort of "super-exactness" condition. Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]