Here is a technical/pedagogical question which has been bothering me for about twelve years.
In problem 5 on page 19 of "Categories for the Working Mathematician" (CWM), Saunders Mac Lane points out that a natural transformation may be fully extended to an intertwining function from one category to another, intertwining meaning (except in the void case), that the function transforms on one side according to its domain functor and on the other according to its codomain functor. Then on page 42 Mac Lane introduces what he calls "horizontal" composition diagramatically and without reference to the fully extended intertwining functions. But the function composite of such a pair of functions trivially intertwines the function composite of the domain functors with that of the codomain functors, and this function composition operation is very quickly verified to be "horizontal" composition when written in terms of restrictions to sets of objects. Thus Mac Lane and everyone else I have read leaves the impression that an honest verification of, say, the associativity of "horizontal" composition would require a somewhat involved diagrammatic demonstration which, in fact, would be nothing other than the hard way to prove the associativity of function composition. Presumably this has been noticed for a long, long time, but the 1998 edition of CWM did not mention it, and I can't help but be struck by the fact that other authors' terminologies leave the impression that they don't know or don't care that "horizontal", star or Godement composition is function composition.[...]
At least in the book "Elemente der Kategorientheorie" by D. Pumpl\"un the above characterization of natural maps is used explicitely; there is also a short discussion on obtaining simpler proofs this way. For the above reason \circ is used for the "horizontal composition"; \cdot or \ast (I do not remember which one) is used for the "vertical composition", which after all looks more "point-wise". Unfortunately some authors use these symbols just the other way round. Marc