 With no further qualification such a category is considered an abstract category. The categories we have been working with are concrete in the sense that they come with a given underlying set or forgetful functor \(U: C\rightarrow\)Set. That is, algebras are based on sets, homomorphisms are certain functions between these sets, and \(U\) simply “forgets” the algebraic structure. Such forgetful functors are faithful in the sense that for any two morphisms \(f, g: a\rightarrow b\) of \(C\), if \(U(f) = U(g)\) then \(f = g,\) i.e. \(U\) does not identify distinct homomorphisms. In general a concrete category is defined as a category \(C\) together with a faithful forgetful functor \(U: C\rightarrow \textrm{Set}\).