Definition:Contravariant Hom Functor

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Let $\mathbf{Set}$ be the category of sets.

Let $\mathbf C$ be a locally small category.

Let $C \in \mathbf C_0$ be an object of $\mathbf C$.

The contravariant hom functor based at $C$:

$\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$

is the covariant functor defined by:

Object functor:    \(\ds \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \)      
Morphism functor:    \(\ds \operatorname{Hom}_{\mathbf C} \left({f, C}\right): \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \to \operatorname{Hom}_{\mathbf C} \left({A, C}\right), g \mapsto g \circ f \)      for $f: A \to B$

where $\operatorname{Hom}_{\mathbf C} \left({B, C}\right)$ denotes a hom set.

Thus, the morphism functor is defined to be precomposition.

Also known as

Some sources call a hom functor a representable functor.

Also denoted as

All notations for hom classes can be seen for hom functors too.

A contravariant hom functor can also be denoted $h_x$; see the Yoneda embedding.

Also see