Functors and quantifiers There is both a covariant and contravariant power set functor, and . The covariant functor is defined more simply. as the functor which sends a set to and a morphism (here, a function between sets) to the image morphism. That is, for . Elsewhere in this article, the power set was defined as the set of functions of into the set with 2 elements. Formally, this defines a natural isomorphism . The contravariant power set functor is different from the covariant version in that it sends to the preimage morphism, so that if . This is because a general functor takes a morphism to precomposition by h, so a function , which takes morphisms from b to c and takes them to morphisms from a to c, through b via h.
Functors and quantifiers There is both a covariant and contravariant power set functor, and . The covariant functor is defined more simply. as the functor which sends a set to and a morphism (here, a function between, sets) to the image morphism. That is, for . Elsewhere in this article, the power set was defined as the set of functions of into the set with 2 elements. Formally, this defines a natural isomorphism . The contravariant power set functor is different from the covariant version in that it sends to the preimage morphism, so that if . This is because a general functor takes a morphism to precomposition by h, so a function  which takes morphisms from b to c and takes them to morphisms from a to c, through b via h.