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Fm(F)g Fq(\()p Fl(B)5 b Fq(\))p 1897 2525 4 351 v 1898 2525 a Fi(?)1707 2366 y Fj(F)i Fg(\()p Fh(h)p Fg(\))1783 3106 y Fm(F)j Fq(\()p Fl(A)p Fq(\))p 1897 3013 V 1898 3013 a Fi(?)1709 2854 y Fj(F)d Fg(\()p Fh(k)r Fg(\))1780 3593 y Fm(F)j Fq(\()p Fl(B)5 b Fq(\))p 1897 3500 V 1898 3500 a Fi(?)1707 3341 y Fj(F)i Fg(\()p Fh(h)p Fg(\))1967 2257 y Fi(@)2001 2291 y(@)-83 b(R)p 2092 2886 4 585 v 2093 2886 a(?)2132 2601 y Fg(1)2167 2615 y Fe(F)5 b Fd(\()p Ff(A)p Fd(\))2001 2988 y Fi(\000)1967 3022 y(\000)-83 b(\011)2015 2745 y(@)2049 2779 y(@)g(R)p 2141 3373 V 2142 3373 a(?)2181 3089 y Fg(1)2216 3103 y Fe(F)5 b Fd(\()p Ff(B)r Fd(\))2049 3476 y Fi(\000)2015 3510 y(\000)-83 b(\011)2504 2127 y Fl(x)2507 2605 y(y)p 2531 2525 4 351 v 2532 2525 a Fi(?)p 2531 2199 4 26 v 2532 2199 a(?)p 2531 3013 4 351 v 2532 3013 a(?)p 2531 2687 4 26 v 2532 2687 a(?)2504 3102 y Fl(x)2507 3580 y(y)p 2531 3500 4 351 v 2532 3500 a Fi(?)p 2531 3174 4 26 v 2532 3174 a(?)149 3779 y Fn(Examples)37 b(8.8.3.)49 b Fq(1.)69 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a Fi(-)1742 4275 y Fh(x)1683 4545 y(y)1605 4445 y Fi(@)1689 4528 y(@)1772 4611 y(@)1855 4694 y(@)1873 4713 y(@)-83 b(R)1961 4815 y Fl(M)5 b(:)p 2023 4713 4 351 v 2025 4713 a Fi(?)2064 4554 y Fh(f)0 4983 y Fq(2.)55 b(Giv)m(en)37 b(mo)s(dules)e Fl(M)898 4998 y Fh(R)993 4983 y Fq(and)1187 4998 y Fh(R)1244 4983 y Fl(N)10 b Fq(.)56 b(De\014ne)38 b Fm(F)44 b Fq(:)34 b Fn(Ab)h Fm(\000)-60 b(!)34 b Fn(Set)j Fq(b)m(y)g Fm(F)10 b Fq(\()p Fl(A)p Fq(\))34 b(:=)h(Bil)3091 4998 y Fh(R)3149 4983 y Fq(\()p Fl(M)5 b(;)17 b(N)10 b Fq(;)17 b Fl(A)p Fq(\).)0 5099 y(Then)32 b Fm(F)40 b Fq(is)31 b(a)f(co)m(v)-5 b(arian)m(t)31 b(functor.)43 b(A)31 b(represen)m(ting)h(ob)5 b(ject)32 b(for)e Fm(F)40 b Fq(is)31 b(giv)m(en)g(b)m(y)h(\()p Fl(M)d Fm(\012)3264 5114 y Fh(R)3341 5099 y Fl(N)5 b(;)17 b Fm(\012)28 b Fq(:)0 5216 y Fl(M)i Fm(\002)20 b Fl(N)38 b Fm(\000)-60 b(!)28 b Fl(M)i Fm(\012)683 5231 y Fh(R)761 5216 y Fl(N)10 b Fq(\))32 b(with)e(the)i(prop)s(ert)m(y)g(that)f(for)g (all)e(\()p Fl(A;)17 b(f)38 b Fq(:)28 b Fl(M)i Fm(\002)20 b Fl(N)38 b Fm(\000)-60 b(!)28 b Fl(A)p Fq(\))j(there)h(exists)p eop %%Page: 38 3 38 2 bop 0 236 a Fo(38)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fq(a)32 b(unique)h Fl(g)e Fm(2)d Fq(Hom\()p Fl(M)k Fm(\012)1015 440 y Fh(R)1096 425 y Fl(N)5 b(;)17 b(A)p Fq(\))32 b(suc)m(h)i(that)f Fm(F)10 b Fq(\()p Fl(g)t Fq(\)\()p Fm(\012)p Fq(\))26 b(=)i(Bil)2413 440 y Fh(R)2471 425 y Fq(\()p Fl(M)5 b(;)17 b(N)10 b Fq(;)17 b Fl(g)t Fq(\)\()p Fm(\012)p Fq(\))27 b(=)h Fl(g)t Fm(\012)f Fq(=)h Fl(f)1384 652 y(M)33 b Fm(\002)23 b Fl(N)154 b(M)33 b Fm(\012)2047 667 y Fh(R)2127 652 y Fl(N)p 1728 627 86 4 v 1731 625 a Fi(-)1743 599 y Fj(\012)1684 885 y Fh(f)1610 776 y Fi(@)1693 859 y(@)1776 942 y(@)1859 1025 y(@)1878 1044 y(@)-83 b(R)1979 1146 y Fl(A:)p 2027 1044 4 351 v 2029 1044 a Fi(?)2068 877 y Fh(g)0 1329 y Fq(3.)41 b(Giv)m(en)27 b(a)g Fc(K)18 b Fq(-mo)s(dule)31 b Fl(V)22 b Fq(.)41 b(De\014ne)28 b Fm(F)37 b Fq(:)28 b Fn(Alg)e Fm(\000)-60 b(!)28 b Fn(Set)f Fq(b)m(y)h Fm(F)10 b Fq(\()p Fl(A)p Fq(\))27 b(:=)h(Hom)o(\()p Fl(V)5 b(;)17 b(A)p Fq(\).)42 b(Then)28 b Fm(F)37 b Fq(is)26 b(a)0 1445 y(co)m(v)-5 b(arian)m(t)26 b(functor.)42 b(A)27 b(represen)m(ting)h(ob)5 b(ject)27 b(for)g Fm(F)36 b Fq(is)26 b(giv)m(en)h(b)m(y)h(\()p Fl(T)14 b Fq(\()p Fl(V)21 b Fq(\))p Fl(;)c(\023)28 b Fq(:)g Fl(V)49 b Fm(\000)-60 b(!)27 b Fl(T)14 b Fq(\()p Fl(V)22 b Fq(\)\))k(with)0 1561 y(the)39 b(prop)s(ert)m(y)g(that)f(for) g(all)e(\()p Fl(A;)17 b(f)48 b Fq(:)38 b Fl(V)59 b Fm(\000)-60 b(!)37 b Fl(A)p Fq(\))i(the)g(exists)g(a)f(unique)h Fl(g)h Fm(2)e Fq(Mor)3053 1576 y Fb(Alg)3182 1561 y Fq(\()p Fl(T)14 b Fq(\()p Fl(V)21 b Fq(\))p Fl(;)c(A)p Fq(\))0 1678 y(suc)m(h)34 b(that)e Fm(F)10 b Fq(\()p Fl(g)t Fq(\)\()p Fl(\023)p Fq(\))27 b(=)h(Hom)o(\()p Fl(V)5 b(;)17 b(g)t Fq(\)\()p Fl(\023)p Fq(\))27 b(=)h Fl(g)t(\023)f Fq(=)h Fl(f)1480 1886 y(V)358 b(T)14 b Fq(\()p Fl(V)21 b Fq(\))p 1588 1862 278 4 v 1783 1860 a Fi(-)1714 1842 y Fh(\023)1662 2121 y(f)1588 2012 y Fi(@)1671 2095 y(@)1754 2178 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b(uniqueness)i(w)m(e)f(get)f Fl(f)38 b Fq(=)28 b Fl(g)t Fq(.)42 b(Hence)31 b(all)d Fl(')p Fq(\()p Fl(B)5 b Fq(\))29 b(are)h(bijectiv)m(e)g(with)f(in)m(v)m(erse)j (map)0 5216 y Fl( )t Fq(\()p Fl(B)5 b Fq(\).)43 b(It)33 b(is)f(su\016cien)m(t)i(to)e(sho)m(w)i(that)e Fl( )37 b Fq(is)32 b(a)g(natural)g(transformation.)p eop %%Page: 39 4 39 3 bop 1169 236 a Fo(8.)77 b(REPRESENT)-6 b(ABLE)33 b(FUNCTORS)1065 b(39)149 425 y Fq(Giv)m(en)33 b Fl(g)e Fq(:)d Fl(B)k Fm(\000)-60 b(!)28 b Fl(C)7 b Fq(.)44 b(Then)33 b(the)g(follo)m(wing)d(diagram)h(comm)m(utes)1159 1092 y(Mor)1335 1107 y Fj(C)1380 1092 y Fq(\()p Fl(A;)17 b(C)7 b Fq(\))612 b Fm(F)10 b Fq(\()p Fl(C)d Fq(\))p 1678 1069 555 4 v 2150 1067 a Fi(-)1876 1136 y Fh( )r Fg(\()p Fh(C)e Fg(\))1158 604 y Fq(Mor)1334 619 y Fj(C)1379 604 y Fq(\()p Fl(A;)17 b(B)5 b Fq(\))610 b Fm(F)10 b Fq(\()p Fl(B)5 b Fq(\))p 1679 581 553 4 v 2149 579 a Fi(-)1875 543 y Fh( )r Fg(\()p Fh(B)s Fg(\))p 1403 999 4 351 v 1404 999 a Fi(?)1034 840 y Fg(Mor)1162 851 y Fe(C)1202 840 y Fg(\()p Fh(A;g)r Fg(\))p 2377 999 V 2379 999 a Fi(?)2418 840 y Fj(F)i Fg(\()p Fh(g)r Fg(\))0 1276 y Fq(since)33 b Fl( )t Fq(\()p Fl(C)7 b Fq(\)Mor)635 1291 y Fj(C)680 1276 y Fq(\()p Fl(A;)17 b(g)t Fq(\)\()p Fl(f)11 b Fq(\))26 b(=)i Fl( )t Fq(\()p Fl(C)7 b Fq(\)\()p Fl(g)t(f)k Fq(\))26 b(=)i Fm(F)10 b Fq(\()p Fl(g)t(f)h Fq(\)\()p Fl(a)p Fq(\))26 b(=)h Fm(F)10 b Fq(\()p Fl(g)t Fq(\))p Fm(F)g Fq(\()p Fl(f)h Fq(\)\()p Fl(a)p Fq(\))26 b(=)h Fm(F)10 b Fq(\()p Fl(g)t Fq(\))p Fl( )t Fq(\()p Fl(B)5 b Fq(\)\()p Fl(f)11 b Fq(\))p Fl(:)149 1393 y Fm(\()p Fq(:)31 b(Let)g Fl(A)h Fq(b)s(e)f(giv)m(en.)43 b(Let)31 b Fl(a)d Fq(:=)g Fl(')p Fq(\()p Fl(A)p Fq(\))1604 1356 y Fj(\000)p Fg(1)1698 1393 y Fq(\(1)1785 1408 y Fh(A)1842 1393 y Fq(\).)42 b(F)-8 b(or)31 b Fl(y)g Fm(2)d(F)10 b Fq(\()p Fl(B)5 b Fq(\))30 b(w)m(e)i(get)f Fl(y)g Fq(=)d Fl(')p Fq(\()p Fl(B)5 b Fq(\))3268 1356 y Fj(\000)p Fg(1)3362 1393 y Fq(\()p Fl(f)11 b Fq(\))27 b(=)0 1509 y Fl(')p Fq(\()p Fl(B)5 b Fq(\))219 1473 y 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