Definition:Contravariant Hom Functor

Definition

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 $C$:

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

		Object functor:		$\displaystyle \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) $
		Morphism functor:		$\displaystyle \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.

This article is complete as far as it goes, but it could do with expansion, in particular:
And this is inconvenient because of concept overloading, but we haven't covered the other type of representable functor
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Also denoted as

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

Although this article appears correct, it's inelegant. There has to be a better way of doing it.
The above should be explicated
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A contravariant hom functor can also be denoted $h_x$; see the Yoneda embedding.

Also see

Definition:Covariant Hom Functor

Contravariant Hom Functor is Functor, where it is shown that $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right)$ is a functor.

Sources

2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 5.5$

Definition:Contravariant Hom Functor

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Definition

Also known as

Also denoted as

Also see

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