Most categories considered in everyday life are concrete; examples are the category of topological spaces with continuous maps as morphisms or the category of groups with group homomorphisms as morphisms.
If C is a concrete category, then there exists a forgetful functor F : C → Set which assigns to every object of C the underlying set and to every morphism in C the corresponding function. This functor is faithful[?], i.e. it maps different morphisms between the same objects to different functions (it may however map different objects to the same set). In the formal approach, a concrete category is defined as a category together with a faithful functor into the category of sets.
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