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.
It is in Section 5 of Peter Freyd Remarks on Algebraically Compact Categories, LMS LNS 177, 1992. (modulo initial invariant = initial algebra). The full dinaturality of initial algebra delivery (as a diagram of functors) is in Section 4 of Adam Eppendahl Coalgebra-to-algebra Morphisms, ENTCS 29, 1999. where it is seen to follow from the lemma: If p is a coalgebra for GF and s is an algebra for FG, then morphisms from Fp to s correspond one-for-one to morphisms from p to Gs (even without an adjunction between G and F). Adam Eppendahl