is_isomorphic X Y := nonempty (X ≅ Y) is an equivalence relation on the objects of a category. The quotient with respect to this relation defines a functor from our category to Type.
An object X is isomorphic to an object Y, if X ≅ Y is not empty.
is_isomorphic defines a setoid.
The functor that sends each category to the quotient space of its objects up to an isomorphism.