Is there a term for the converse of "unique up to isomorphism"? Many people say that an object satisfying a property is unique up to isomorphism if every such object belongs to a unique isomorphism equivalence class. Is there a term for an object that is both unique up to isomorphism and also satisfies the converse: if an object is isomorphic to an object with this property, then this object also has the property?
For most of the objects we study in category theory, this if and only if holds. One can, however, create artificial examples where the converse does not hold.
 A: "Invariant under isomorphism." 
A: There is the term "topological invariant": Any two topological spaces homeomorphic to this one specified space that have property $X$.  That is expressed by saying $X$ is a topological invariant.  One reason this term gets used is that things initially defined in terms of metric properties of a space unexpectedly turn out to be topological invariants.  Thus the integral of Gaussian curvature with respect to area of a compact surface without boundary requires more than topological ideas for its definition, but every pair of such manifolds that are homeomorphic to each other have the same integral of Gaussian curvature with respect to area.
I don't know of any examples for other sorts of isomorphism that seem as good as things like that.
A: The object is "unique up to arbitrary isomorphism".  Which is to say that
The property "defines the object up to arbitrary isomorphisms".  Which is to say that 
The property defines [identifies, specifies, cuts out, assigns, etc] a unique isomorphism class of objects.
The last formulation seems like the most standard use of language, and makes it clear that isomorphism could be replaced by any equivalence relation. 
