Full and faithful functors

In category theory, a faithful functor (resp. a full functor) is a functor that is injective (resp. surjective) when restricted to each set of morphisms that have a given source and target.

Formal definitions

Explicitly, let C and D be (locally small) categories and let F : CD be a functor from C to D. The functor F induces a function

for every pair of objects X and Y in C. The functor F is said to be

for each X and Y in C.


A faithful functor need not be injective on objects or morphisms. That is, two objects X and X may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : XY and f : XY (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C.

A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : CD is a full and faithful functor and then .


See also


  1. Mac Lane (1971), p. 15
  2. 1 2 Jacobson (2009), p. 22
  3. Mac Lane (1971), p. 14


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