Locally small category

A category is locally small if between any pair of objects there is only a set's worth of morphisms.

It is traditional to write 𝒞(X,Y) or Hom(X,Y) for the set of morphisms from X to Y in a locally small category 𝒞.

The set of arrows between a pair of fixed objects in a locally small category is typically called a hom-set, whether or not it is a set of homomorphisms of any particular kind.

Because the notation is so convenient, it is also adopted for the collection of morphisms between a fixed pair of objects in a category that is not necessarily locally small.