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Thus, F is faithful, and by symmetry, so is G. 10, Gk and the isomorphisms ηc and ηc define a unique h : c → c for which both Gk and GFh make the diagram c ηc Gk or GFh h c G GFc ηc G GFc 38The reader is strongly encouraged to stop reading here and attempt to prove this result on their own. 10. 32 1. CATEGORIES, FUNCTORS, NATURAL TRANSFORMATIONS commute. 10 again, GFh = Gk, whence Fh = k by faithfulness of G. Thus, F is full, faithful, and essentially surjective. For the converse, suppose now that F : C → D is full, faithful, and essentially surjective on objects.
What is a functor between groups, regarded as one-object categories? ii. What is a functor between preorders, regarded as categories? iii. Find an example to show that the objects and morphisms in the image of a functor F : C → D do not necessarily define a subcategory of D. iv. 11 are functorial. v. What is the difference between a functor Cop → D and a functor C → Dop ? What is the difference between a functor C → D and a functor Cop → Dop ? vi. , so that f · Fh = Gk · f . Define a pair of projection functors dom : F ↓G → D and cod : F ↓G → E.
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. A full and faithful functor, called fully faithful for short, that is injective-on-objects defines a full embedding of the domain category into the codomain category.