Dear Gaunce Lewis, GT-colleague and all, When we regard a monoid M as a one-object category, an M-set is a functor X : M --> Set and the colimit of the functor X is the set of orbits of the M-set. What GT-colleague has is an ordered monoid which can be regarded as a one-object 2-category M, and the action F of M on the category C amounts to a 2-functor X : M --> Cat. I suspect that the construction required is the pseudocolimit of X. This kind of colimit for 2-functors was considered in the book of John Gray J.W. Gray, Formal Category Theory: Adjointness for 2-Categories, Lecture Notes in Math. 391 (Springer, 1974) and in my paper Limits indexed by category-valued 2-functors, J. Pure Appl. Algebra 8 (1976) 149-181 where I show that pseudo(co)limits and lax (co)limits are ordinary weighted (= indexed) (co)limits in the sense of enriched category theory (for the base monoidal category Cat). I suspect the condition that the identity element is initial is a red herring even though this makes it look as though canonical maps are being inverted rather than isomorphisms being introduced. Regards, Ross