I have a homework problem asking me "which kind of functors preserve split coequalisers?" - I have seen multiple online sources (such as the comments in http://golem.ph.utexas.edu/category/2007/04/schur_functors.html and Pierre Antoine Grillet's Abstract Algebra) claiming that "all functors preserve split coequalisers"): however, suppose our coequaliser h is a coequaliser for f and g (so hf=hg), and a functor sends f and g to the same morphism: F(f)=F(g). Then surely our coequaliser would no longer be a coequaliser?
I don't expect you to provide the full answer to the question if you don't want, just to correct the flaw in my logic if there is one! Surely a coequaliser needs 2 morphisms on which to be a coequaliser. [Incidentally, I believe my 'split coequaliser' follows the standard category theory definition.]
Thankyou very much - Ben