In topology, there are some properties that are not topological, i.e. that are not preserved by homeomorphisms. Think for instance of completeness or of the property of being a Cauchy sequence.
Now, in category theory, we can use the terminology categorical properties to indicate all those properties preserved by isomorphisms of categories (I'm not interested in equivalence but in isomorphisms). Think for instance of the repleteness of a subcategory, its reflectivity and so on. Now, my question is: are there some properties that are not categorical (in the above sense)?