Actually, F isn't even a functor. The unique arrow 0 --> Ub has to give a canonical arrow F0 = 1 --> b, which there isn't. You could choose one, of course, but it could not be functorial. Actually, I realized the answer to my question cannot be yes. Here's why. Let A be some complete category to be specified later. Let d be a fixed object of A. Let B be set\op and Fa = Hom(a,d). The right adjoint is given by b |---> d^b. It is not entirely trivial to show this, but if my answer were "yes", then you could show that the class of objects that were equalizers of powers of d would be complete. It is obviously closed under products but, over 40 years ago, Isbell gave an example in which it was not closed under equalizers. This much is true: if there is an equalizer of the form a --> UFa ===> Ub, then a ---> UFa ===> UFUFa is an equalizer. Michael On Tue, 18 Mar 2008, Vaughan Pratt wrote:

Isn't the following a counterexample?

Let A = Set and let B = A\{0} (the category of nonempty sets). Let F send the empty set in A to the singleton set in B, and otherwise let F and U be the evident identity functors between A and B. Similarly let \eta and \epsilon be the identity natural transformations, except for \eta_0 which can only be the unique function from 0 to 1. Naturality of \eta and \epsilon depends on both being the identity, except for \eta_0 but that's from the initial object so all its diagrams commute.

Then 0 equalizes the two arrows from U1 to U2 but \eta_0 does not equalize UF\eta a and \eta UFa since the latter two are both 1_1 in A whence they are equalized by 1.

Vaughan

Michael Barr wrote:

...

I guess I am getting old and dumb. This question should have been a snap for me years ago. It is old fashioned, only a 1-categorical question and not about internal vs. external.

Suppose F: A --> B is left adjoint to U: B --> A. Suppose a is an object of A and b, b' objects of B such that there is an equalizer a ---> Ub ===> Ub'. (The two arrows Ub to UB' are not assumed to be U of arrows from B.) Does it follow that a ---> UFa ===> UFUFa is an equalizer? The arrows are \eta a, UF\eta a and \eta UFa of course.

Michael