Dear Paul, I proved a "topical" version of this as Propn 2.3.7 in my "Topical Categories of Domains" (1999). "Topical" here means working in the 2-category of Grothendieck toposes and geometric morphisms instead of that of categories and functors. Instead of objects of a category and morphisms between them, it deals with points of a topos and natural transformations between them. (Note that I use the term "F-structures" instead of "F- algebras".) In this setting there are some subtleties of interpretation. An initial F-structure is defined as a point of the classifying topos [F- struct] for F-structures that is initial amongst all the generalized points - making [F-struct] a local topos. Nonetheless, the argument is essentially one that you might use with categories and functors. I remarked that my results were familiar from the category context as set out in Freyd's 1991 paper "Algebraically complete categories". I cannot remember if your result on FG-algebras and GF-algebras was in Freyd. All the best, Steve. On 28 Jan 2009, at 19:31, Paul Levy wrote:

Does anybody know a reference for the following (very easy) result?

Let C and D be categories, and let F:C-->D and G:D-->C be functors.

If (c,theta) is an initial algebra for GF, then (Fc, F theta) is an initial algebra for FG.

thanks, Paul