We consider equality atoms 
, i.e., is a countable set with the equality relation. Recall that by definable set we mean a structure which can be defined (in a possibly nested fashion) from using finite unions, finite tuples, and set-builder expressions with first-order formulas ranging over , e.g. 
is definable using the parameter 
. A definable relational structure is a relational structure 
where 
and each relation 
is a definable set. Up to isomorphism, definable structures are the same as structures which interpret in , using first-order interpretations (this correspondence preserves parameters), but sometimes using definable sets makes constructions easier.
The problem described in the previous post is to determine isomorphism between two definable structures. In this post, we describe this problem in terms of automorphism groups. This post is based on discussions with Manuel Bodirsky and Antoine Mottet.
-categorical structure 
, the pure countable set.
and 
are isomorphic, No otherwise.