# Definition:Hom Bifunctor

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## Definition

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

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

The **hom bifunctor** on $\mathbf C$ is the covariant functor $\map {\operatorname {Hom} } {-, -} : \mathbf C^{\operatorname {op} } \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets such that:

- $(1): \quad \map {\operatorname {Hom} } {a, b}$ is the hom class

- $(2): \quad$ If $\tuple {f^{\operatorname {op} }, g}: \tuple {a, b} \to \tuple {c, d}$ is a morphism, $\map {\operatorname {Hom} } {f^{\operatorname{op} }, g}: \map {\operatorname {Hom} } {a, b} \to \map {\operatorname {Hom} } {c, d}$ is $f_* \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.

## Also denoted as

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