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4) that a bijective semigroup homomorphism must be an isomorphism. That is the same as saying that the underlying functor from Sem to Set reflects isomorphisms. The same remark applies to Mon. The underlying functor from the category of posets and monotone maps does not reflect isomorphisms. A full and faithful functor reflects isomorphisms, but in fact it does a bit more than that, as described by the following proposition. 10 Proposition Let F : C − → D be full and faithful, and suppose A and B are objects of C and u : F (A) − → F (B) is an isomorphism in D.

Precisely, we define a function φ∗ : A∗ × S − FA–1 φ∗ ((), s) = s for s ∈ S. FA–2 φ∗ ((a)w, s) = φ(a, φ∗ (w, s)) for any s ∈ S, w ∈ A∗ and a ∈ A. Recall that the free monoid F (A) is the set A∗ with concatenation as multiplication. The function φ as just defined is thus an action of F (A) on S. The identity of A∗ is the empty word () and by FA–1, φ∗ ((), a) = a for all a ∈ A, so A–1 follows. As for A–2, if we assume that ∗ φ∗ (wv, m) = φ∗ (w, φ∗ (v, m)) for words w of length k, then φ∗ ((a)wv, m) = φ(a, φ∗ (wv, m)) = φ(a, φ∗ (w, φ∗ (v, m))) = φ∗ ((a)w, φ∗ (v, m)) The first and third equality are from the definition of φ, while the second is from the inductive hypothesis.

10, we defined the function Hom(C, f ) : Hom(C, A) − → Hom(C, B) by setting Hom(C, f )(g) = f ◦ g for every g ∈ Hom(C, A), that is for g : C − → A. We use this function to define the covariant hom functor Hom(C, −) : C − → Set as follows: HF–1 Hom(C, −)(A) = Hom(C, A) for each object A of C ; HF–2 Hom(C, −)(f ) = Hom(C, f ) : Hom(C, A) − → Hom(C, B) for f : A − → B. The following calculations show that Hom(C, −) is a functor. For an object A, Hom(C, idA ) : Hom(C, A) − → Hom(C, A) takes an arrow f : C − → A to idA ◦ f = f ; hence Hom(C, idA ) = idHom(C,A) .

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