By Barr M.
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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) .