Questions on dinatural transformations.

["Noson Yanofsky" <noson@sci.brooklyn.cuny.edu>]

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