I understand the definition and usefulness of the notion of functor.
But I am worrying about the usefulness of the notion of a contravariant functor. Wikipedia writes:
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor [...]
But why do they "turn morphisms around", wouldn't it be easier to do the same without the inversion of morphisms and composition?
So I guess it would be beneficial for me to know some examples of naturally occuring contravariant functors. So let me ask: what are some constructions in mathematics that naturally occur as contravariant functors instead of covariant functor?