Re: Function composition of natural transformations? (Pat Don...

[Jpdonaly@aol.com]

Thanks, Jean. As an assiduous student of CWM, I was aware of this and will
always wonder why Mac Lane didn't just make the point explicit in his first
edition in 1971. The only thing left to realize is that the category of
commutative squares which you mention is a subcategory of a product category and thus
has a couple of projection functors on it which can be used to follow a functor
to get the domain and codomain functors of the natural transformation, so that
this version of naturality is much more neatly packaged than the usual
diagram. I believe that there is a worker named John Baez (deep apologies for any
naive and unforgivable errors here) who says that Mac Lane claimed to be
interested not in functoriality so much as naturality when he was coinventing
category theory; I wonder when and if he realized that naturality is a brand of
functoriality. It would seem that this realization would come very early. In
general, if one fixes an argument in a bifunctor, the resulting function is a fully
extended intertwining function, and I believe that your point is that every
natural transformation arises in this way. So already naturality is an artifact
of functoriality. Mitchell notices much of this in his 1965 book.