Further to my question on adjoints

[Michael Barr <barr@math.mcgill.ca>]

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