From: noson@sci.brooklyn.cuny.edu I was hoping that the category of small categories, functors and dinat transformations...
There's a category problem already at this point. Dinats don't go between functors F,G:C->D, they go between sesquifunctors F:C^op x C->D and differ from n.t.'s of that type by only being defined on the diagonal of C^op x C. The off-diagonal and non-identity-morphism entries in F,G only participate in the dinaturality condition, not in the transformation itself.
a) It is well known that there is no vertical composition of dinatural transformations. How about horizontal composition?
Before you can compose dinats horizontally you have to be able to compose the sesquifunctors they bridge. I don't know how others do this, but if I had to compose G:D^op x D -> E with F:C^op x C -> D, my inclination would be to restrict the evident composite G(F(a,b),F(c,d)) to a=d, b=c (i.e. where the variances match up). That is, GoF:C^op x C -> E is defined by G(F(c,c'),F(c',c)) on object pairs (c',c) of C^op x C, with the expected extension to morphism pairs (f',f) where f':c'->d' in C^op (i.e. f':d'->c' in C) and f:c->d in C, namely G(F(f,f'),F(f',f)): G(F(c,c'),F(c',c)) -> G(F(d,d'),F(d',d)). With that (or some) choice of sesquifunctor composition one can then ask about horizontal composition tos where s:F->F', t:G->G'. How would you whisker a dinatural on the left, i.e. apply the whisker G:D^op x D->E on the left to the dinat s:F->F' on the right where F,F':C^op x C->D? For natural transformations, G is just a functor G:D->E, so this is just a matter of applying G pointwise to each s_c. For dinaturals however, G is a sesquifunctor. What do you want a sesquifunctor to do to a morphism s_c? Maybe there's some span-like thing one can do here but I don't see it. For dinaturals, vertical composition may turn out to be easier than horizontal, in that it at least makes sense provided one solves the shape-matching problem somehow. In doing so one also solves another problem, that dinaturality is too weak a condition, typically admitting transformations on the internal hom that aren't Church numerals (Pare & Roman, JPAA 128 33-92 for Set, Pratt, TCS 294:3, bottom of p461, for Chu(Set,K) and chu(Set,K) which awkwardly seem to need different treatments). Mike Barr has a notion of strong dinatural (unpublished?), and the notion of binary (more generally n-ary) logical transformation also works well here when definable on the category of interest. Vaughan Pratt