Saying without examples is a pedagogical nonsense, I can't argue, because I agree.

But if I provide the two typical examples Hom(A,) and Home(,A) and how they transport the arrows it starts to click.

There is also List which is a covariant functor. The stupid Home(unit,) which is the "identity" functor.

I could also add Hom(Bool,) which is the pair functor, ie Hom(Bool,A) is a tuple in AxA. A->AxA (or A->Him(Bool, A)) is a covariant functor.

Etc. Examples are the keys to understand the idea.