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Ii) If f : x → y and g : y → z are morphisms so that g f is monic, then f is monic. Dually: (i’) If f : x y and g : y z are epimorphisms, then so is g f : x z. (ii’) If f : x → y and g : y → z are morphisms so that g f is epic, then g is epic. Proof. iii. Exercises. i. Show that C/c (c/(Cop ))op . iii(i). ii. (i) Show that a morphism f : x → y is a split epimorphism in a category C if and only if for all c ∈ C, post-composition f∗ : C(c, x) → C(c, y) defines a surjective function. (ii) Argue by duality that f is a split monomorphism if and only if for all c ∈ C, pre-composition f ∗ : C(y, c) → C(x, c) is a surjective function.

7. 5. i). 5. EQUIVALENCE OF CATEGORIES 31 • and essentially surjective on objects if for every object d ∈ D there is some c ∈ C such that d is isomorphic to Fc. 8. ” A faithful functor need not be injective on morphisms; neither must a full functor be surjective on morphisms. A faithful functor that is injective on objects is called an embedding and identifies the domain category as a subcategory of the codomain; in this case, faithfulness implies that the functor is (globally) injective on arrows.

I) For vector spaces of any dimension, the map ev : V → V ∗∗ that sends v ∈ V to the linear function evv : V ∗ → k defines the components of a natural transformation from the identity endofunctor on Vectk to the double dual functor. To check that the naturality square ev G ∗∗ V V φ  W φ∗∗ ev  G W ∗∗ commutes for any linear map φ : V → W, it suffices to consider the image of a generic vector v ∈ V. By definition, evφv : W ∗ → k carries a functional f : W → k to f (φv). 7(ii) on morphisms, we see that φ∗∗ (evv ) : W ∗ → k carries a functional f : W → k to f φ(v), which amounts to the same thing.