Faithful functor

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

Explicitly, let C and D be (locally small) categories and let F : C → D 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 : X → Y and f′ : X′ → Y′ (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 : C → D is a full and faithful functor and ${\displaystyle F(X)\cong F(Y)}$ then ${\displaystyle X\cong Y}$.

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