Re: Neil Ghani's question

Mike Barr's answer to Neil Ghani's question is very pretty, but unfortunately wrong. There are many examples of cartesian natural transformations (e.g. between functors Set --> Set) which are neither epic nor monic: for example, the natural transformation (-) x A --> (-) x B induced by an arbitrary map A --> B. The mistake in Mike's proof occurs when he says

Let E be the equalizer of u and v and let h^B ---> E be any arrow.

The trouble is that E could be the zero functor, so that there may not be any arrows h^B --> E. Peter Johnstone