Full and faithful functors

Summary

A functor is faithful when it is injective on each set of morphisms between two fixed objects. It is full when it is surjective on each such set, and fully faithful when it is both. A faithful functor loses no information about morphisms, and a fully faithful one identifies the morphisms of its source with those between the images. Forgetful functors such as groups to sets are faithful but not full. Fully faithful functors are the embeddings of category theory, and the Yoneda embedding is the main example. Together with essential surjectivity they characterize equivalences of categories.