By Max Kelly

Initially released as: Cambridge collage Press, Lecture Notes in arithmetic sixty four, 1982

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**Example text**

3 below. 3 The isomorphism [A ⊗ B, C] ∼ = [A, [B, C]] We return to the case where [A, B] exists. 21) /G G′ : C /G [A, B] to the composite E(β ⊗ 1A ). 20) and sending β : G is 2-natural in C; and in fact it is an isomorphism of categories. 21) for a To prove this we must ﬁrst show that every P : C ⊗ A unique G. 21) gives P (C, −) = GC, P (−, A) = EA G. 23) ) B P (C, A), P (D, A) . 8(c), this determines a unique GCD by the universal property of EA . 6) for G follow at once from those for P (−, A).

30) expresses F K as the end ∫ [A(K, A), F A], so that we have an isomorphism A ϕ: FK ∼ = [A, V](A(K, −), F ]. 8(m), is the composite /G [X, F A] A(K, A) F /G [F K, F A] KA 1 [η,1] For a more useful citation, see Peter Freyd and Ross Street, On the size of categories, Theory and Applications of Categories 1 (1995) 174-178. 48). This is for a unique η : X equivalent to the assertion that αA = ϕA η for a unique η; which completes the proof. 31) is a bijection The image under V = V0 (I, −) : V0 V0 (I, F K) /G [A, V]0 (A(K, −), F ).

For a general V it would take us into side-issues to discuss the various kinds of monomorphisms and epimorphisms in the context of V-categories; so we refrain from deﬁning faithfulness for a V-functor, and hence from deﬁning a generator for a V-category. 8 below are needed to justify its proof in that case; in the meantime we use it only for V = Set. 40 The inclusion Z : A generating if B is the closure of A for some set Φ of colimits. 6 Strongly generating functors 47 generating, B is the closure of A for small colimits; provided that B is cocomplete, that B0 is ﬁnitely complete, and that each object of B0 has only a small set of extremal-epimorphic quotients.