\begin{definition}[Definition:Category]
A '''category''' is an interpretation of the metacategory axioms within set theory.
Because a metacategory is a metagraph, this means that a '''category''' is a graph.
Let $\mathfrak U$ be a class of sets.
A metacategory $\mathbf C$ is a '''category''' {{iff}}:
:$(1): \quad$ The objects form a subset $\mathbf C_0$ or $\operatorname {ob} \ \mathbf C \subseteq \mathfrak U$
:$(2): \quad$ The morphisms form a subset $\mathbf C_1$ or $\operatorname{mor} \ \mathbf C$ or $\operatorname{Hom} \ \mathbf C \subseteq \mathfrak U$
A '''category''' is what one needs to define in order to define a functor.
\end{definition}