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: \circ g^*$, the postcomposition with $g$ composed with the precomposition with $f$.
 * $\operatorname{Hom}(a, b)$ 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_*

Also denoted as
All notations for hom classes can be seen for hom functors too.

Also see

 * Definition:Local Hom Bifunctor