# Definition:Hom Bifunctor

## 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 $\operatorname{Hom}(-, -) : \mathbf C^{\operatorname{op}} \times \mathbf C \to \mathbf {Set}$ from the product with the opposite category to the category of sets with:

- $\operatorname{Hom}(a, b)$ is the hom class
- If $(f^{\operatorname{op}}, g) : (a, b) \to (c, d)$ is a morphism, $\operatorname{Hom}(f^{\operatorname{op}}, g) : \operatorname{Hom}(a, b) \to \operatorname{Hom}(c, d)$ 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.