is there a property of a category that is preserved by category isomorphism, but not equivalence? I am trying to teach myself category theory and I am reading the section for Equivalence of categories. After some reading the question of "if there is a property of a category that is preserved by category isomorphism, but not equivalence" came to mi mind. The definitions are kind of similar and I simply cannot come up with an example about it. 
any help will be appreciated since I cannot concentrate or move into the reading since my mind cannot rest due to the question above. 
 A: Sure, the property of being a set for example.  A category is called a set if the only morphisms are identity morphisms.  Categories which are sets can be equivalent to categories which are not sets.
Also, the property of being skeletal.  A category is skeletal if no two distinct objects are isomorphic.  Most categories you work with are not skeletal, but every category is equivalent to a skeletal category.  In fact, that skeletal category is unique up to isomorphism, so two categories are equivalent if and only if they have isomorphic skeletal subcategories.
Edit: For example, let $A$ be the category with one object $\operatorname{obj}(A) = \{\ast\}$ and only the identity morphism.  Let $B$ be the category with two objects $\operatorname{obj}(B) = \{1, 2\}$ and whose only morphisms are the identities and maps $f\colon 1 \to 2$ and $g\colon 2 \to 1$ satisfying $fg = \operatorname{id}_2$ and $gf = \operatorname{id}_1$.
Simply put, $A$ is a category with one object and only the identity maps, $B$ is a category with two objects and only the identity maps and the isomorphisms between those objects.  Then $A$ and $B$ are equivalent.  I'll let you figure out the functors.  Pretty much anything you write down that is actually a functor is going to work.
A: The following are invariant under isomorphisms but not under equivalences: 
The size of the class of objects. 
The homeomorphism class of the classifying space of the category.
Having a unique initial object. More generally, having unique solutions to a given universal problem is such a property. (very close to Jim's answer about skeletality.)
The group of automorphisms of the category, that is the group of isomorphisms $F:C\to C$ under composition. 
The cardinality of the longest path of non-repeating arrows.
In some sense, the first two are the most important such properties. The first since it's very elementary, the second since it relates categories and isomorphisms to topology, and categories and equivalences to homotopy theory. The remaining invariants are very artificial. This is to be expected, as any property of categories that is worth mentioning should be invariant under equivalences.
