Covariant Hom functor

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Definition

Suppose is a category and . The covariant Hom functor corresponding to is defined as a covariant functor from to the category of sets given as follows:

  • On objects: An object is mapped to .
  • On morphisms: Given objects and an element , gets sent to the map defined by:

.

A set-valued functor is termed representable if it is naturally isomorphic to a covariant Hom functor.

Facts