A functor is said to be faithful if it is injective on maps. This 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 injective on maps. For example, a counterexample
is the functor on the category
of vector spaces which leaves every vector space
unchanged and sends every map to the zero map.

A functor which is injective both on objects and maps is sometimes called an embedding.