In other words, a functor F : C → D is faithful if the maps
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 : X → Y and f′ : X′ → Y′ may map to the same morphism in D.
For example, the forgetful functorU : Grp → Set is faithful but neither injective on objects or morphisms.
Two-place functors that are covariant in both arguments are called bifunctors.
As a rule, a construction that may be defined for any object of a category or for any sequence of objects of a fixed length, independently of the individual properties of the objects, is likely to be functorial.
is called faithful if these mappings are all injective, and full if they are all surjective.
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