Examples
- The forgetful functor U : Grp → Set is faithful as each group maps to a unique set and the group homomorphism are a subset of the functions. This functor is not full as there are functions between groups which are not group homomorphisms. A category with a faithful functor to Set is (by definition) a concrete category; in general, that forgetful functor is not full.
- Let F : Set → Set be the functor which maps every set to the empty set and every function to the empty function. Then F is full, but is neither injective on objects nor on morphisms.
- The inclusion functor Ab → Grp is fully faithful.
Read more about this topic: Full And Faithful Functors
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