\begin{definition}[Definition:Graph (Category Theory)]
A '''graph''' is an interpretation of a metagraph within set theory.
Let $\mathfrak U$ be a class of sets.
A metagraph $\GG$ is a '''graph''' {{iff}}:
:$(1): \quad$ The objects form a subset $\operatorname{vert} \GG \subseteq \mathfrak U$
:$(2): \quad$ The morphisms form a subset $\operatorname{edge} \GG \subseteq \mathfrak U$
If the class $\mathfrak U$ is a set, then morphisms are functions, and the domain and codomain in the definition of a morphism are those familiar from set theory.
If $\mathfrak U$ is a proper class this is not the case, for example the morphisms of $\CC$ need not be functions.
Category:Definitions/Category Theory
\end{definition}