In general, the naive horizontal merging of dinaturals fails to be dinatural. This is discussed in the article "Functorial Polymorphism" by Bainbridge, Freyd, Scedrov and me (Theoretical Computer Science 1990, pp. 35-64). Several counterexamples are given there. For example, in a cartesian closed category of domains or CPO's, consider a dinatural family Y_A: A^A --> A (e.g. in domains, let Y_A = the least fixed point operator). If you were able to compose this with the "polymorphic identity" dinat id_A: 1---> A^A (i.e. a dinat from constant functor 1 to (-) ==> (-) where id_A = the transpose of the identity on A), then the category would be degenerate (proved in BFSS, Appendix A.4). Of course, if the middle diamond (of an attempted merging of two dinat families) is a pullback or pushout, then merging works. (see BFSS, Fact 1.2). Re vertical merging, some things can be said quite generally: e.g BFSS, Propn. 1.3. For various generalizations, see Peter Freyd's paper "Structural Polymorphism" (in TCS, 1993, pp.107-129). Soloviev has also discussed compositionality of dinats in several articles in JPAA. Philip Scott On Tue, 29 Jun 2004, Noson Yanofsky wrote:
Hello,
Two quick questions:
a) It is well known that there is no vertical composition of dinatural transformations. How about horizontal composition?
i.e. Given S,S':C^op x C---->B T,T':C^op x C ----> B^op U,U':B^op x B --->A \alpha: S--->S' dinat \alpha': T--->T' dinat and \beta: U--->U' dinat
is there a \beta \circ (\alpha',\alpha) and is it dinat? It should be. But I can not seem to find the right definition.
How about if we restrict to a nice category of moduals for a nice algebra over a nice field? Does that help?
I was hoping that the category of small categories, functors and dinat transformations should be a graph-category (a category enriched over graphs) but am having a hard time finding what the composition is. Did someone write on these things?
b) Also, I was wondering if anyone ever wrote about quasi-dinatural transformations. Those are dinats where the target category is a 2-category and the hexagon commutes up to a two cell. They show up in something I am working on. But they are very painful. Has anyone worked on such things?
Any thoughts?
All the best, Noson Yanofsky