Locally small category
From AoPSWiki
A locally small category is a category whose hom-sets are (small) sets. More explicitly, it is a category such that for all objects 
and 
in the category, there exists a set 
whose elements are exactly the morphisms from to .
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 and , there does exist a set that contains all the morphisms from to , i.e., all the functions 
.
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