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