Suppose $F:C\to D$ is a left adjoint. Let $U:{\mathsf{Cat}}\to {\mathsf{Ab}}$ be the left adjoint to the fully faithful functor ${\mathsf{Ab}}\to {\mathsf{Cat}}$ that views an abelian group as a category with one-object. If $U(C)$ is isomorphic to $U(D)$, is $C$ isomorphic to $D$?

This question probably has something to do with the 2-categorial structure on ${\mathsf{Cat}}$ and ${\mathsf{Ab}}$ and whether $U$ preserves this structure. However I'm not too familiar with 2-category theory.

someleft adjoint functor $C \to D$? Or do you actually mean the following: If $U(F) : U(C) \to U(D)$ is an isomorphism, is the same true for $F$? Have you checked some examples? I doubt that anything along these lines may be true ... – Martin Brandenburg Jun 18 '12 at 14:59