Consider the dual finitary question: In universal algebra in order to show that finitely presented objects are closed under coequalizers it is essential that a amorphism of finitely presented objects lift to a morphism between the free. Is this the only way to prove it ? : " but when I look at examples, it has turned out to be true for other reasons." greetings e.d.
In March I asked a question on adjoints, to which I have received no correct response. Rather than ask it again, I will pose what seems to be a simpler and maybe more manageable question. Suppose C is a complete category and E is an object. Form the full subcategory of C whose objects are equalizers of two arrows between powers of E. Is that category closed in C under equalizers? (Not, to be clear, the somewhat different question whether it is internally complete.)
In that form, it seems almost impossible to believe that it is, but it is surprisingly hard to find an example. When E is injective, the result is relatively easy, but when I look at examples, it has turned out to be true for other reasons. Probably there is someone out there who already knows an example.
Michael