Let be a category. A subcategory of is given by
*a subcollection of objects of , denoted ,
*a subcollection of morphisms of , denoted .such that
*for every in , the identity morphism id is in ,
*for every morphism in , both the source and the target are in ,
*for every pair of morphisms and in the composite is in whenever it is defined.

These conditions ensure that is a category in its own right: its collection of objects is , its collection of morphisms is , and its identities and