Definition:Locally Small Category

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Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is said to be locally small if and only if all of its hom classes are sets.

That is, if and only if for all objects $X, Y \in \mathbf C_0$ of $\mathbf C$:

$\operatorname{Hom}_{\mathbf C} \left({X, Y}\right) = \set {f \in \mathbf C_1: f: X \to Y}$

is a set.