hold:
** for every object in C,
** for all morphisms and in C.

That is, functors must preserve identity morphisms and composition of morphisms.

Covariance and contravariance

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 F from C to D as a mapping that
* associates each object

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