non-unital monads

Hello, I am trying to find some information about non-unital monads (gadgets with \mu but without \eta). In particular I am interested in the following two questions: 1. Given a non-unital monad can it have two different "unitality" structures? 2. Is there a concept of a free non-unital monad? For example, I can think of the "free" non-unital monad generated by the functor X |-> X^2 on sets as the monad that sends a set X into the set of "homogeneous" expressions made with one binary operation s such that there is s(x1,x2) and s(s(x1,x2),s(x3,x4)) but no x1 itself and no s(x1,s(x2,x3)). But what is the universal characterization of it? Thanks! Vladimir. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]