Definition:Category

Definition
A category is an interpretation of the metacategory axioms within set theory.

Universal Definition
Let $\mathbb U$ be a Grothendieck universe of sets.

A metacategory $\mathcal C$ is a category if:


 * 1. The objects form a subset $\mathcal C_0$ or $\operatorname{ob}\mathcal C \subseteq \mathbb U$


 * 2. The morphisms form a subset $\mathcal C_1$ or $\operatorname{mor}\mathcal C$ or $\operatorname{Hom}\mathcal C \subseteq \mathbb U$

With this definition the morphisms are functions, and the domain, codomain and composition in the definition of a metacategory are those familiar from set theory.

Definition Using Classes
The definition of a category using classes is identical to the above definition, except that the universe $\mathbb U$ is replaced with a class $\mathcal A$.

Using class comprehension, the morphisms of $\mathcal C$ need not be functions.