A functor 
is faithful if, for every pair of objects 
, the induced function
 |
is an injection. Thus distinct morphisms between the same pair of objects remain distinct after
applying 
.
A fully faithful functor requires each of
these functions to be a bijection,
so every fully faithful functor is faithful, but a faithful functor need not be full.
Faithfulness does not necessarily imply injectivity on objects. For example, the forgetful functor from the category of groups to the category of sets is faithful, but it identifies non-isomorphic groups having the same underlying set. Conversely, a functor injective on objects need not be faithful. For example, regard a nontrivial group as a category with one object and its group elements as morphisms. The functor to the category with one object and only its identity morphism is injective on objects but sends every group element to the same morphism.
A functor which is injective both on objects and maps is sometimes called an embedding.
