A question on adjoints

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