Subcategory
Formal definition

Let C be a category. A subcategory S of C is given by

such that

These conditions ensure that S is a category in its own right: its collection of objects is ob(S), its collection of morphisms is hom(S), and its identities and composition are as in C. There is an obvious faithful functor I : S → C, called the inclusion functor which takes objects and morphisms to themselves.

Let S be a subcategory of a category C. We say that S is a full subcategory of C if for each pair of objects X and Y of S,

A full subcategory is one that includes all morphisms in C between objects of S. For any collection of objects A in C, there is a unique full subcategory of C whose objects are those in A.
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