# Faithful functor

In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target.

In other words, a functor F : CD is faithful if the maps

[itex]F_{X,Y}:\mathrm{Mor}_{\mathcal C}(X,Y)\rightarrow\mathrm{Mor}_{\mathcal D}(FX,FY)[itex]

are injective for every pair of objects X and Y in C.

Note that 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, and two morphisms f : XY and f′ : X′ → Y′ may map to the same morphism in D.

For example, the forgetful functor U : GrpSet is faithful but neither injective on objects or morphisms.

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