Hom functor
Properties

Note that a functor of the form

is a presheaf; likewise, Hom(A, –) is a copresheaf.

A functor F : C → Set that is naturally isomorphic to Hom(A, –) for some A in C is called a representable functor (or representable copresheaf); likewise, a contravariant functor equivalent to Hom(–, A) might be called corepresentable.

Note that Hom(–, –) : C × C → Set is a profunctor, and, specifically, it is the identity profunctor id C: C ↛ C {\displaystyle \operatorname {id} _{C}\colon C\nrightarrow C}.

The internal hom functor preserves limits; that is, hom (X, −): C → C {\displaystyle \operatorname {hom} (X,-)\colon C\to C} sends limits to limits, while hom (−, X): C op → C {\displaystyle \operatorname {hom} (-,X)\colon C^{\text{op}}\to C} sends limits in C op {\displaystyle C^{\text{op}}}, that is colimits in C {\displaystyle C}, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.

The endofunctor Hom(E, –) : Set → Set can be given the structure of a monad; this monad is called the environment (or reader) monad.
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