Definition:Graph (Category Theory)

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A graph is an interpretation of a metagraph within set theory.

Let $\mathfrak U$ be a class of sets.

A metagraph $\mathcal G$ is a graph if:

$(1): \quad$ The objects form a subset $\operatorname{vert}\mathcal G \subseteq \mathfrak U$
$(2): \quad$ The morphisms form a subset $\operatorname{edge}\mathcal G \subseteq \mathfrak U$

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 $\mathcal C$ need not be functions.