Graham, I'm not sure if anyone mentioned the examples of a poset and a monoid as categories. There is no "internal" structure in these. In the former, the objects are the elements and there is a morphism between a and b iff a <= b. A functor then becomes an order preserving map. In the latter, there is one object and the morphisms are the elements. The identity is the identity map and if x and y are two elements / morphisms then composition is xy. A functor is then a homomorphism. Dominic.