Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows.
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.
Some authors define an embedding to be a full and faithful functor that is injective on objects.
[1] Other authors define a functor to be an embedding if it is faithful and injective on objects.
With the definitions of the previous paragraph, for any (full) embedding F : B → C the image of F is a (full) subcategory S of C, and F induces an isomorphism of categories between B and S. If F is not injective on objects then the image of F is equivalent to B.