A note on a category composition

The special properties of an abstract category morphism (for instance, being an identity, an isomorphism, an epimorphism., a monomorphism . . . ) fully depend on the category composition. Consequently, an isomorphic category to a concrete category may be not concrete, i.e., the concreteness is not a category invariant. Further, every small category is isomorphic to a small category whose objects are sets and whose morphisms are functions between those sets.