representable functor

Given a category C, a covariant representable functor h sends each object X in category C to C(A,X), the set of arrows from A to X, and sends each arrow f:X → Y to a function from C(A,X) to C(A,Y) which maps each element s of C(A,X) to an element of C(A,Y). A contravariant representable functor h sends each object X in category C to set C(X,A) and each arrow f:X → Y to a function from C(Y,A) to C(X,A) which maps each element s of C(Y,A) to an element of C(X,A). Note: h:C → Sets and h:C → Sets, because h reverses the direction of the morphisms, whereas h does not.

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