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By Bodo Pareigis

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Proof. 14. T h e functor i n Corollary 1 denoted by 0 will be called the evaluation functor. N o w we want to apply the new results for representable functors. S 44 1. COROLLARY 2. (a) (b) PRELIMINARY NOTIONS Let A,Be h e Mor (h , h ) is a bijection. or<#(A, B) and the natural isomorphisms in Mor (h , h ). F : ^ —> S, we have M o r ( A ^ , #~) ^ f B A f (c) / &(A). (d) Mor<^(^4, JB) a / h->• hj e Morj(h , h ) is a bijection, inducing a bijection between the isomorphisms in Mor<#(A, B) and the natural isomorphisms in Mor (h , h ).

I n ? the distinguished points of Mor^(^4, B) are uniquely determined by the condition that M o r ^ ( / , —) and Mor^(-yg) are pointed set maps. I n the category S an initial object is 0 and a final object is {0}. Zero objects do not exist. T h e only zero morphisms have the form 0 —> A. I n the category S* each set with one point is a zero object. T h u s there are zero morphisms between all objects. Similarly, the set with one point y c 24 1. PRELIMINARY NOTIONS with the corresponding structure is a zero object i n the categories A b , and Top*.

D) | jMor() r Mor (ß C D ) D) M r ( ° ^ / , j P ) > M o r ^ C , D) and i f one computes the image of l ^ . T h e converse is trivial. 5—the isomorphism has to be tested only argumentwise— and from Corollary 2(b). W e define an equivalence relation on the class of objects i n the following way. T w o objects are called equivalent if the representable functors, represented by these objects, are isomorphic. B y the Yoneda lemma this is the same equivalence relation as the one defined by isomorphisms of objects.

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