Definition:Hom Class

Definition
Let $\mathbf C$ be a metacategory.

Let $C$ and $D$ be objects of $\mathbf C$.

The collection of morphisms $f: C \to D$ is called a hom class and is denoted $\map {\operatorname {Hom}_{\mathbf C} } {C, D}$.

Also known as
If $\map {\operatorname {Hom}_{\mathbf C} } {C, D}$ is a set, then it is also called a hom set.

Some authors hyphenate, resulting in hom-class and hom-set.

Also denoted as
When the category $\mathbf C$ is clear, it is mostly dropped from the notation, yielding $\map {\operatorname {Hom} } {C, D}$.

The hom class is also denoted $\map {\mathbf C} {C, D}$, or in the case of a functor category, $\map {\operatorname {Nat} } {C, D}$.

Also see

 * Definition:Hom Functor