(a) The forgetful functor M → S (see the Example preceding Definition 1.2) is representable.

(b) The forgetful functor G → S is representable.

Data from definition 1.2

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Definition 1.2. Let C and D be categories. A contravariant functor S from C to D (denoted S: CD) is a pair of functions (both denoted by S), an object function which assigns to each object C ofe an object S(C) of D and a morphism function which as- signs to each morphism f: C→C' of a morphism S(f): S(C)→ S(C) of D such that (i) S(1c) = 1s(c) for every identity morphism 1c of C; (ii) S(gof) = S(f) S(g) for any two morphisms f, g of e whose composite g of is defined. Thus the morphism function of a contravariant functor S: CD reverses the direction of morphisms. EXAMPLE. Let R be a ring and B a fixed left R-module. Define a contravariant functor S from the category of left R-modules to the category of abelian groups by defining S(C) = Hom(C,B) for each R-module C. Iff: C→C' is an R-module homomorphism, then S() is the induced map f: Hom(C',B)→ Hom(C,B) (see the Remarks after Theorem IV.4.1). EXAMPLE. More generally, let B be a fixed object in a category C. Define a con- travariant functor h³ from C to the category S of sets by assigning to each object C of e the set hB(C) = hom(C,B) of all morphisms in e from C to B. If f: C → C' is a morphism of C, let (f): hom(C',B)→ hom(C,B) be the function given by ggof (g a hom(C',B)). The functor h is called the contravariant hom functor.