Locally small category

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A locally small category is a category whose hom-sets are (small) sets. More explicitly, it is a category such that for all objects $A$ and $B$ in the category, there exists a set $\text{Hom}(A,B)$ whose elements are exactly the morphisms from $A$ to $B$.

Most categories encountered outside category theory are locally small. For example, the category of sets is a locally small category, even though it is not a small category. This is because there is no (small) set containing all (small) sets, but for any two sets $A$ and $B$, there does exist a set $\text{Hom}(A,B)$ that contains all the morphisms from $A$ to $B$, i.e., all the functions $f : A \to B$.

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