Embeddings
Given a subcategory S of C the inclusion functor I : S → C is both faithful and injective on objects. It is full if and only if S is a full subcategory.
Many authors define an embedding to be a full and faithful functor.
Other authors define a functor to be an embedding if it is faithful and injective on objects. Equivalently, F is an embedding if it is injective on morphisms. A functor F is then called a full embedding if it is a full functor and an embedding.
For any (full) embedding F : B → C the image of F is a (full) subcategory S of C and F induces a isomorphism of categories between B and S.
In some categories, one can also speak of morphisms of the category being embeddings.
Read more about this topic: Subcategory