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.



If the universal abelian group of two adjoint categories are isomoprhic, are the original two categories isomorphic? 
The universal abelian group for a finite poset is the free abelian group on the edges of the Hasse diagram. Consider the posets $P=\lbrace 0,a,b,1\rbrace$ with $0\lneq a,b\lneq 1$ and $a,b$ incomparable and $Q=\lbrace 0,1,2,3,4\rbrace$ with the usual order. Then the universal group in both cases is then free abelian of rank 4. Consider the map $F\colon P\to Q$ sending $0,a,b$ to $0$ and $1$ to $4$. This is an order-preserving function between complete lattices preserving infs and so has a left adjoint by the adjoint functor theorem which I am too lazy to write down. The posets are not isomorphic.


**Added.**


The left adjoint seems to be $G(0) = 0$ and $G(j) = 1$ for $j=1,2,3,4$. 


