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Fn(\))p 2148 1513 V 2150 1513 a Fg(?)2189 1363 y Fj(G)6 b Fn(\()p Fh(f)11 b Fn(\))0 1893 y(comm)m(utes)32 b(for)h(all)d Fh(f)38 b Fn(:)28 b Fh(A)g Fj(\000)-60 b(!)28 b Fh(B)37 b Fn(in)32 b Fj(C)6 b Fn(,)33 b(i.e.)43 b Fj(G)6 b Fn(\()p Fh(f)11 b Fn(\))p Fh(')p Fn(\()p Fh(A)p Fn(\))27 b(=)h Fh(')p Fn(\()p Fh(B)5 b Fn(\))p Fj(F)10 b Fn(\()p Fh(f)h Fn(\).)149 2107 y Fk(Lemma)38 b(8.3.2.)49 b Fi(Given)g(c)-5 b(ovariant)49 b(functors)h Fj(F)65 b Fn(=)55 b(Id)2350 2122 y Ff(Set)2524 2107 y Fn(:)g Fk(Set)g Fj(\000)-58 b(!)56 b Fk(Set)50 b Fi(and)f Fj(G)61 b Fn(=)0 2223 y(Mor)176 2238 y Ff(Set)294 2223 y Fn(\(Mor)508 2238 y Ff(Set)627 2223 y Fn(\()p Fj(\000)p Fh(;)17 b(A)p Fn(\))p Fh(;)g(A)p Fn(\))27 b(:)h Fk(Set)g Fj(\000)-58 b(!)28 b Fk(Set)35 b Fi(for)f(a)h(set)g Fh(A)p Fi(.)45 b(Then)34 b Fh(')28 b Fn(:)f Fj(F)38 b(\000)-58 b(!)28 b(G)41 b Fi(with)605 2449 y Fh(')p Fn(\()p Fh(B)5 b Fn(\))28 b(:)g Fh(B)33 b Fj(3)28 b Fh(b)g Fj(7!)f Fn(\(Mor)1518 2464 y Ff(Set)1636 2449 y Fn(\()p Fh(B)5 b(;)17 b(A)p Fn(\))28 b Fj(3)g Fh(f)38 b Fj(7!)28 b Fh(f)11 b Fn(\()p Fh(b)p Fn(\))27 b Fj(2)i Fh(A)p Fn(\))e Fj(2)h(G)6 b Fn(\()p Fh(B)f Fn(\))0 2675 y Fi(is)35 b(a)f(natur)-5 b(al)35 b(tr)-5 b(ansformation.)149 2889 y Fe(Pr)n(oof.)42 b Fn(Giv)m(en)33 b Fh(g)e Fn(:)d Fh(B)k Fj(\000)-60 b(!)28 b Fh(C)7 b Fn(.)44 b(Then)33 b(the)g(follo)m(wing)d(diagram)h(comm)m(utes)786 3170 y Fh(B)414 b Fn(Mor)1450 3185 y Ff(Set)1568 3170 y Fn(\(Mor)1782 3185 y Ff(Set)1901 3170 y Fn(\()p Fh(B)5 b(;)17 b(A)p Fn(\))p Fh(;)g(A)p Fn(\))p 894 3147 351 4 v 1162 3145 a Fg(-)960 3101 y Fh(')p Fn(\()p Fh(B)5 b Fn(\))787 3658 y Fh(C)421 b Fn(Mor)1454 3673 y Ff(Set)1572 3658 y Fn(\(Mor)1786 3673 y Ff(Set)1904 3658 y Fn(\()p Fh(C)r(;)17 b(A)p Fn(\))p Fh(;)g(A)p Fn(\))p 893 3635 356 4 v 1166 3633 a Fg(-)962 3589 y Fh(')p Fn(\()p Fh(C)7 b Fn(\))p 824 3565 4 351 v 826 3565 a Fg(?)736 3401 y Fh(g)p 1799 3565 V 1800 3565 a Fg(?)1839 3414 y Fn(Mor)2016 3429 y Ff(Set)2134 3414 y Fn(\(Mor)2348 3429 y Ff(Set)2466 3414 y Fn(\()p Fh(g)t(;)17 b(A)p Fn(\))p Fh(;)g(A)p Fn(\))0 3867 y(since)632 4066 y Fh(')p Fn(\()p Fh(C)7 b Fn(\))p Fj(F)j Fn(\()p Fh(g)t Fn(\)\()p Fh(b)p Fn(\)\()p Fh(f)h Fn(\))26 b(=)h Fh(')p Fn(\()p Fh(C)7 b Fn(\))p Fh(g)t Fn(\()p Fh(b)p Fn(\)\()p Fh(f)k Fn(\))27 b(=)g Fh(f)11 b(g)t Fn(\()p Fh(b)p Fn(\))28 b(=)f Fh(')p Fn(\()p Fh(B)5 b Fn(\)\()p Fh(b)p Fn(\)\()p Fh(f)11 b(g)t Fn(\))366 4182 y(=)28 b([)p Fh(')p Fn(\()p Fh(B)5 b Fn(\)\()p Fh(b)p Fn(\)Mor)1009 4197 y Ff(Set)1127 4182 y Fn(\()p Fh(g)t(;)17 b(A)p Fn(\)]\()p Fh(f)11 b Fn(\))27 b(=)g([Mor)1867 4197 y Ff(Set)1985 4182 y Fn(\(Mor)2199 4197 y Ff(Set)2317 4182 y Fn(\()p Fh(g)t(;)17 b(A)p Fn(\))p Fh(;)g(A)p Fn(\))p Fh(')p Fn(\()p Fh(A)p Fn(\)\()p Fh(b)p Fn(\)]\()p Fh(f)11 b Fn(\))p Fh(:)p 3529 4401 4 66 v 3533 4339 59 4 v 3533 4401 V 3591 4401 4 66 v 149 4615 a Fk(Lemma)38 b(8.3.3.)49 b Fi(L)-5 b(et)25 b Fh(f)39 b Fn(:)27 b Fh(A)h Fj(\000)-58 b(!)28 b Fh(B)i Fi(b)-5 b(e)25 b(a)g(morphism)e(in)i Fj(C)6 b Fi(.)41 b(Then)25 b Fn(Mor)2797 4630 y Fd(C)2842 4615 y Fn(\()p Fh(f)5 b(;)17 b Fi(-)p Fn(\))27 b(:)h(Mor)3308 4630 y Fd(C)3353 4615 y Fn(\()p Fh(B)5 b(;)17 b Fi(-)p Fn(\))27 b Fj(\000)-58 b(!)0 4731 y Fn(Mor)176 4746 y Fd(C)221 4731 y Fn(\()p Fh(A;)17 b Fi(-)p Fn(\))40 b Fi(given)f(by)i Fn(Mor)1057 4746 y Fd(C)1102 4731 y Fn(\()p Fh(f)5 b(;)17 b(C)7 b Fn(\))37 b(:)h(Mor)1630 4746 y Fd(C)1675 4731 y Fn(\()p Fh(B)5 b(;)17 b(C)7 b Fn(\))38 b Fj(3)g Fh(g)j Fj(7!)c Fh(g)t(f)48 b Fj(2)38 b Fn(Mor)2745 4746 y Fd(C)2790 4731 y Fn(\()p Fh(A;)17 b(C)7 b Fn(\))40 b Fi(is)g(a)g(natur)-5 b(al)0 4848 y(tr)g(ansformation)34 b(of)h(c)-5 b(ovariant)34 b(functors.)149 4964 y(L)-5 b(et)36 b Fh(f)k Fn(:)29 b Fh(A)h Fj(\000)-58 b(!)29 b Fh(B)41 b Fi(b)-5 b(e)35 b(a)h(morphism)e(in)h Fj(C)6 b Fi(.)47 b(Then)35 b Fn(Mor)2175 4979 y Fd(C)2220 4964 y Fn(\()p Fi(-)p Fh(;)17 b(f)11 b Fn(\))29 b(:)g(Mor)2695 4979 y Fd(C)2740 4964 y Fn(\()p Fi(-)p Fh(;)17 b(A)p Fn(\))29 b Fj(\000)-58 b(!)29 b Fn(Mor)3321 4979 y Fd(C)3366 4964 y Fn(\()p Fi(-)p Fh(;)17 b(B)5 b Fn(\))0 5080 y Fi(given)24 b(by)i Fn(Mor)538 5095 y Fd(C)583 5080 y Fn(\()p Fh(C)r(;)17 b(f)11 b Fn(\))26 b(:)i(Mor)1091 5095 y Fd(C)1136 5080 y Fn(\()p Fh(C)r(;)17 b(A)p Fn(\))27 b Fj(3)h Fh(g)j Fj(7!)c Fh(f)11 b(g)31 b Fj(2)d Fn(Mor)2134 5095 y Fd(C)2179 5080 y Fn(\()p Fh(C)r(;)17 b(B)5 b Fn(\))25 b Fi(is)g(a)g(natur)-5 b(al)25 b(tr)-5 b(ansformation)0 5196 y(of)35 b(c)-5 b(ontr)g(avariant)34 b(functors.)p eop %%Page: 8 3 8 2 bop 0 236 a Fl(8)1494 b(8.)77 b(TOOLBO)n(X)149 425 y Fe(Pr)n(oof.)42 b Fn(Let)33 b Fh(h)28 b Fn(:)g Fh(C)34 b Fj(\000)-60 b(!)28 b Fh(C)1174 389 y Fd(0)1230 425 y Fn(b)s(e)33 b(a)f(morphism)f(in)h Fj(C)6 b Fn(.)43 b(Then)34 b(the)f(diagrams)1054 1133 y(Mor)1230 1148 y Fd(C)1275 1133 y Fn(\()p Fh(B)5 b(;)17 b(C)1513 1096 y Fd(0)1536 1133 y Fn(\))458 b(Mor)2208 1148 y Fd(C)2253 1133 y Fn(\()p Fh(A;)17 b(C)2485 1096 y Fd(0)2508 1133 y Fn(\))p 1603 1110 401 4 v 1921 1108 a Fg(-)1624 1178 y Fc(Mor)1751 1189 y Fb(C)1792 1178 y Fc(\()p Fa(f)s(;C)1931 1154 y Fb(0)1953 1178 y Fc(\))1066 645 y Fn(Mor)1242 660 y Fd(C)1287 645 y Fn(\()p Fh(B)5 b(;)17 b(C)7 b Fn(\))481 b(Mor)2220 660 y Fd(C)2265 645 y Fn(\()p Fh(A;)17 b(C)7 b Fn(\))p 1591 622 425 4 v 1933 620 a Fg(-)1635 584 y Fc(Mor)1763 595 y Fb(C)1803 584 y Fc(\()p Fa(f)s(;C)e Fc(\))p 1312 1039 4 351 v 1314 1039 a Fg(?)936 881 y Fc(Mor)1063 892 y Fb(C)1103 881 y Fc(\()p Fa(B)s(;h)p Fc(\))p 2287 1039 V 2289 1039 a Fg(?)2328 881 y Fc(Mor)2455 892 y Fb(C)2496 881 y Fc(\()p Fa(A;h)p Fc(\))0 1352 y Fn(and)1068 2000 y(Mor)1244 2015 y Fd(C)1290 2000 y Fn(\()p Fh(C)r(;)17 b(A)p Fn(\))485 b(Mor)2216 2015 y Fd(C)2261 2000 y Fn(\()p Fh(C)r(;)17 b(B)5 b Fn(\))p 1582 1977 429 4 v 1928 1975 a Fg(-)1629 2044 y Fc(Mor)1757 2055 y Fb(C)1797 2044 y Fc(\()p Fa(C)q(;f)i Fc(\))1054 1512 y Fn(Mor)1230 1527 y Fd(C)1275 1512 y Fn(\()p Fh(C)1390 1476 y Fd(0)1413 1512 y Fh(;)17 b(A)p Fn(\))458 b(Mor)2202 1527 y Fd(C)2247 1512 y Fn(\()p Fh(C)2362 1476 y Fd(0)2385 1512 y Fh(;)17 b(B)5 b Fn(\))p 1597 1489 401 4 v 1915 1487 a Fg(-)1616 1452 y Fc(Mor)1744 1463 y Fb(C)1784 1452 y Fc(\()p Fa(C)1866 1429 y Fb(0)1889 1452 y Fa(;f)i Fc(\))p 1309 1907 4 351 v 1311 1907 a Fg(?)936 1748 y Fc(Mor)1064 1759 y Fb(C)1104 1748 y Fc(\()p Fa(h;A)p Fc(\))p 2284 1907 V 2286 1907 a Fg(?)2325 1748 y Fc(Mor)2452 1759 y Fb(C)2493 1748 y Fc(\()p Fa(h;B)s Fc(\))0 2196 y Fn(comm)m(ute.)p 3529 2196 4 66 v 3533 2133 59 4 v 3533 2196 V 3591 2196 4 66 v 149 2394 a Fk(Remark)37 b(8.3.4.)50 b Fn(The)35 b(comp)s(osition)d(of)i(t)m(w)m(o)h(natural)f (transformations)e(is)i(again)f(a)i(nat-)0 2510 y(ural)d (transformation.)41 b(The)33 b(iden)m(tit)m(y)g(id)1545 2525 y Fd(F)1606 2510 y Fn(\()p Fh(A)p Fn(\))28 b(:=)g(1)1963 2525 y Fd(F)7 b Fc(\()p Fa(A)p Fc(\))2164 2510 y Fn(is)32 b(also)g(a)g(natural)f(transformation.)149 2708 y Fk(De\014nition)37 b(8.3.5.)49 b Fn(A)37 b(natural)e(transformation)g Fh(')g Fn(:)g Fj(F)44 b(\000)-60 b(!)35 b(G)43 b Fn(is)37 b(called)e(a)i Fi(natur)-5 b(al)39 b(iso-)0 2824 y(morphism)33 b Fn(if)g(there)h (exists)h(a)f(natural)f(transformation)f Fh( )i Fn(:)d Fj(G)36 b(\000)-60 b(!)30 b(F)44 b Fn(suc)m(h)35 b(that)f Fh(')23 b Fj(\016)g Fh( )35 b Fn(=)30 b(id)3550 2839 y Fd(G)0 2940 y Fn(and)39 b Fh( )31 b Fj(\016)26 b Fh(')38 b Fn(=)h(id)664 2955 y Fd(F)725 2940 y Fn(.)63 b(The)40 b(natural)e(transformation)f Fh( )43 b Fn(is)38 b(uniquely)h (determined)g(b)m(y)h Fh(')p Fn(.)63 b(W)-8 b(e)0 3056 y(write)32 b Fh(')313 3020 y Fd(\000)p Fc(1)435 3056 y Fn(:=)c Fh( )t Fn(.)149 3172 y(A)46 b(functor)g Fj(F)55 b Fn(is)45 b(said)g(to)g(b)s(e)h Fi(isomorphic)e Fn(to)i(a)f(functor)h Fj(G)52 b Fn(if)44 b(there)i(exists)h(a)e(natural)0 3289 y(isomorphism)30 b Fh(')e Fn(:)f Fj(F)38 b(\000)-60 b(!)27 b(G)6 b Fn(.)149 3486 y Fk(Problem)36 b(8.3.1.)49 b Fn(1.)57 b(Let)37 b Fj(F)10 b Fh(;)17 b Fj(G)41 b Fn(:)35 b Fj(C)41 b(\000)-60 b(!)36 b(D)j Fn(b)s(e)f(functors.)57 b(Sho)m(w)38 b(that)f(a)f(natural)g(trans-)0 3603 y(formation)27 b Fh(')h Fn(:)g Fj(F)37 b(\000)-60 b(!)28 b(G)36 b Fn(is)29 b(a)h(natural)e(isomorphism)g(if)g(and)i(only)g(if)f Fh(')p Fn(\()p Fh(A)p Fn(\))g(is)h(an)f(isomorphism)0 3719 y(for)j(all)f(ob)5 b(jects)33 b Fh(A)28 b Fj(2)g(C)6 b Fn(.)149 3835 y(2.)51 b(Let)35 b(\()p Fh(A)24 b Fj(\002)g Fh(B)5 b(;)17 b(p)861 3850 y Fa(A)918 3835 y Fh(;)g(p)1011 3850 y Fa(B)1071 3835 y Fn(\))35 b(b)s(e)g(the)h(pro)s(duct)f(of)g Fh(A)g Fn(and)g Fh(B)40 b Fn(in)34 b Fj(C)6 b Fn(.)51 b(Then)36 b(there)g(is)e(a)h(natural)0 3951 y(isomorphism)909 4108 y(Mor\(-)o Fh(;)17 b(A)22 b Fj(\002)h Fh(B)5 b Fn(\))1539 4081 y Fj(\030)1539 4112 y Fn(=)1644 4108 y(Mor)1820 4123 y Fd(C)1865 4108 y Fn(\(-)o Fh(;)17 b(A)p Fn(\))22 b Fj(\002)h Fn(Mor)2388 4123 y Fd(C)2433 4108 y Fn(\(-)o Fh(;)17 b(B)5 b Fn(\))p Fh(:)149 4286 y Fn(3.)62 b(Let)39 b Fj(C)45 b Fn(b)s(e)39 b(a)g(category)g(with)g(\014nite)f(pro)s (ducts.)63 b(F)-8 b(or)38 b(eac)m(h)i(ob)5 b(ject)39 b Fh(A)g Fn(in)f Fj(C)45 b Fn(sho)m(w)40 b(that)0 4402 y(there)34 b(exists)g(a)f(morphism)e(\001)1139 4417 y Fa(A)1225 4402 y Fn(:)e Fh(A)g Fj(\000)-60 b(!)28 b Fh(A)23 b Fj(\002)g Fh(A)33 b Fn(satisfying)f Fh(p)2314 4417 y Fc(1)2354 4402 y Fn(\001)2435 4417 y Fa(A)2521 4402 y Fn(=)c(1)2674 4417 y Fa(A)2760 4402 y Fn(=)g Fh(p)2913 4417 y Fc(2)2953 4402 y Fn(\001)3034 4417 y Fa(A)3091 4402 y Fn(.)45 b(Sho)m(w)34 b(that)0 4518 y(this)e(de\014nes)j(a)d (natural)f(transformation.)42 b(What)32 b(are)h(the)g(functors?)149 4634 y(4.)87 b(Let)47 b Fj(C)54 b Fn(b)s(e)47 b(a)g(category)g(with)g (\014nite)g(pro)s(ducts.)88 b(Sho)m(w)48 b(that)e(there)i(is)f(a)g(bi)p Fi(functor)0 4751 y Fn(-)22 b Fj(\002)h Fn(-)29 b(:)g Fj(C)g(\002)23 b(C)35 b(\000)-60 b(!)29 b(C)39 b Fn(suc)m(h)c(that)e (\(-)22 b Fj(\002)i Fn(-)o(\)\()p Fh(A;)17 b(B)5 b Fn(\))33 b(is)g(the)h(ob)5 b(ject)34 b(of)f(a)g(pro)s(duct)g(of)g Fh(A)g Fn(and)h Fh(B)5 b Fn(.)46 b(W)-8 b(e)0 4867 y(denote)33 b(elemen)m(ts)f(in)g(the)g(image)f(of)g(this)h(functor)h(b)m(y)g Fh(A)21 b Fj(\002)h Fh(B)33 b Fn(:=)27 b(\(-)21 b Fj(\002)h Fn(-)o(\)\()p Fh(A;)17 b(B)5 b Fn(\))32 b(and)h(similarly)0 4983 y Fh(f)g Fj(\002)22 b Fh(g)t Fn(.)149 5099 y(5.)90 b(With)48 b(the)g(notation)f(of)h(the)g(preceding)h(problem)e(sho)m(w)i (that)f(there)h(is)e(a)h(natural)0 5216 y(transformation)36 b Fh(\013)q Fn(\()p Fh(A;)17 b(B)5 b(;)17 b(C)7 b Fn(\))37 b(:)h(\()p Fh(A)26 b Fj(\002)h Fh(B)5 b Fn(\))26 b Fj(\002)h Fh(C)1831 5188 y Fj(\030)1831 5220 y Fn(=)1946 5216 y Fh(A)f Fj(\002)h Fn(\()p Fh(B)k Fj(\002)c Fh(C)7 b Fn(\).)61 b(Sho)m(w)39 b(that)f(the)h(diagram)p eop %%Page: 9 4 9 3 bop 1137 236 a Fl(3.)77 b(NA)-6 b(TURAL)31 b(TRANSF)n(ORMA)-6 b(TIONS)1070 b(9)0 425 y Fn(\()p Fi(c)-5 b(oher)g(enc)g(e)34 b(or)g(c)-5 b(onstr)g(aints)p Fn(\))-1 610 y(\(\()p Fh(A)23 b Fj(\002)f Fh(B)5 b Fn(\))22 b Fj(\002)h Fh(C)7 b Fn(\))22 b Fj(\002)h Fh(D)635 b Fn(\()p Fh(A)22 b Fj(\002)h Fn(\()p Fh(B)k Fj(\002)22 b Fh(C)7 b Fn(\)\))22 b Fj(\002)h Fh(D)p 858 587 576 4 v 1351 585 a Fg(-)949 549 y Fa(\013)p Fc(\()p Fa(A;B)s(;C)5 b Fc(\))p Fd(\002)p Fc(1)2924 610 y Fh(A)22 b Fj(\002)h Fn(\(\()p Fh(B)k Fj(\002)c Fh(C)7 b Fn(\))22 b Fj(\002)h Fh(D)s Fn(\))p 2320 587 V 2813 585 a Fg(-)2401 549 y Fa(\013)p Fc(\()p Fa(A;B)s Fd(\002)p Fa(C)q(;D)r Fc(\))p 413 907 4 254 v 414 907 a Fg(?)453 796 y Fa(\013)p Fc(\()p Fa(A)p Fd(\002)p Fa(B)s(;C)q(;D)r Fc(\))p 3338 907 V 3339 907 a Fg(?)2903 796 y Fc(1)p Fd(\002)p Fa(\013)p Fc(\()p Fa(B)s(;C)q(;D)r Fc(\))-1 1000 y Fn(\()p Fh(A)g Fj(\002)f Fh(B)5 b Fn(\))22 b Fj(\002)h Fn(\()p Fh(C)29 b Fj(\002)23 b Fh(D)s Fn(\))2094 b Fh(A)22 b Fj(\002)h Fn(\()p Fh(B)k Fj(\002)c Fn(\()p Fh(C)29 b Fj(\002)23 b Fh(D)s Fn(\)\))p 858 977 2038 4 v 2813 975 a Fg(-)1668 939 y Fa(\013)p Fc(\()p Fa(A;B)s(;C)5 b Fd(\002)p Fa(D)r Fc(\))0 1135 y Fn(comm)m(utes.)149 1251 y(6.)77 b(With)44 b(the)g(notation)e(of)i(the)g(preceding)g(problem)f(sho)m(w)i(that)e (there)i(are)f(a)f(natural)0 1367 y(transformations)35 b Fh(\025)p Fn(\()p Fh(A)p Fn(\))f(:)g Fh(E)d Fj(\002)25 b Fh(A)34 b Fj(\000)-60 b(!)35 b Fh(A)h Fn(and)h Fh(\032)p Fn(\()p Fh(A)p Fn(\))d(:)g Fh(A)25 b Fj(\002)g Fh(E)41 b Fj(\000)-60 b(!)34 b Fh(A)i Fn(suc)m(h)i(that)e(the)h(diagram)0 1484 y(\()p Fi(c)-5 b(oher)g(enc)g(e)34 b(or)g(c)-5 b(onstr)g(aints)p Fn(\))940 1668 y(\()p Fh(A)22 b Fj(\002)h Fh(E)6 b Fn(\))22 b Fj(\002)h Fh(B)625 b(A)22 b Fj(\002)h Fn(\()p Fh(E)28 b Fj(\002)23 b Fh(B)5 b Fn(\))p 1518 1645 564 4 v 1999 1643 a Fg(-)1648 1607 y Fa(\013)p Fc(\()p Fa(A;E)t(;B)s Fc(\))1663 2063 y Fh(A)22 b Fj(\002)h Fh(B)1212 1855 y Fa(\032)p Fc(\()p Fa(A)p Fc(\))p Fd(\002)p Fc(1)1317 1767 y Fg(Q)1400 1822 y(Q)1483 1878 y(Q)1566 1933 y(Q)1615 1965 y(Q)-83 b(s)2154 1855 y Fc(1)p Fd(\002)p Fa(\025)p Fc(\()p Fa(B)s Fc(\))2200 1767 y Fg(\021)2117 1822 y(\021)2034 1878 y(\021)1951 1933 y(\021)1902 1965 y(\021)g(+)149 2208 y Fk(De\014nition)37 b(8.3.6.)49 b Fn(Let)32 b Fj(C)38 b Fn(and)31 b Fj(D)k Fn(b)s(e)d(categories.)43 b(A)32 b(co)m(v)-5 b(arian)m(t)31 b(functor)h Fj(F)37 b Fn(:)27 b Fj(C)34 b(\000)-60 b(!)28 b(D)34 b Fn(is)0 2324 y(called)f(an)h Fi(e)-5 b(quivalenc)g(e)35 b(of)h(c)-5 b(ate)g(gories)33 b Fn(if)g(there)i(exists)f(a)g(co)m(v)-5 b(arian)m(t)34 b(functor)g Fj(G)i Fn(:)31 b Fj(D)h(\000)-60 b(!)31 b(C)40 b Fn(and)0 2440 y(natural)31 b(isomorphisms)g Fh(')d Fn(:)f Fj(G)6 b(F)1271 2412 y(\030)1272 2444 y Fn(=)1376 2440 y(Id)1465 2455 y Fd(C)1543 2440 y Fn(and)33 b Fh( )e Fn(:)d Fj(F)10 b(G)2056 2412 y(\030)2057 2444 y Fn(=)2161 2440 y(Id)2251 2455 y Fd(D)2312 2440 y Fn(.)149 2556 y(A)32 b(con)m(tra)m(v)-5 b(arian)m(t)32 b(functor)g Fj(F)37 b Fn(:)28 b Fj(C)34 b(\000)-60 b(!)27 b(D)35 b Fn(is)c(called)g(a)g Fi(duality)k(of)f(c)-5 b(ate)g(gories)31 b Fn(if)g(there)h(exists)0 2672 y(a)42 b(con)m(tra)m(v)-5 b(arian)m(t)41 b(functor)h Fj(G)50 b Fn(:)44 b Fj(D)i(\000)-60 b(!)44 b(C)k Fn(and)43 b(natural)d(isomorphisms)g Fh(')k Fn(:)g Fj(G)6 b(F)3145 2645 y(\030)3146 2677 y Fn(=)3266 2672 y(Id)3356 2687 y Fd(C)3443 2672 y Fn(and)0 2789 y Fh( )32 b Fn(:)27 b Fj(F)10 b(G)324 2761 y(\030)325 2793 y Fn(=)429 2789 y(Id)518 2804 y Fd(D)579 2789 y Fn(.)149 2905 y(A)29 b(category)f Fj(C)34 b Fn(is)28 b(said)f(to)h(b)s(e)g Fi(e)-5 b(quivalent)28 b Fn(to)f(a)h(category)h Fj(D)h Fn(if)d(there)i(exists)g(an)f(equiv)-5 b(alence)0 3021 y Fj(F)38 b Fn(:)29 b Fj(C)34 b(\000)-60 b(!)29 b(D)s Fn(.)44 b(A)34 b(category)f Fj(C)39 b Fn(is)33 b(said)f(to)h(b)s(e)g Fi(dual)g Fn(to)g(a)g(category)g Fj(D)j Fn(if)c(there)h(exists)h(a)f(dualit)m(y)0 3137 y Fj(F)k Fn(:)28 b Fj(C)34 b(\000)-60 b(!)28 b(D)s Fn(.)149 3312 y Fk(Problem)36 b(8.3.2.)49 b Fn(1.)44 b(Sho)m(w)33 b(that)f(the)h(dual)f(category)h Fj(C)2337 3276 y Fa(op)2444 3312 y Fn(is)f(dual)g(to)g(the)h(category)g Fj(C)6 b Fn(.)149 3428 y(2.)46 b(Let)33 b Fj(D)j Fn(b)s(e)e(a)f(category)g(dual) g(to)g(the)g(category)h Fj(C)6 b Fn(.)46 b(Sho)m(w)34 b(that)f Fj(D)j Fn(is)d(equiv)-5 b(alen)m(t)33 b(to)g(the)0 3544 y(dual)f(category)h Fj(C)668 3508 y Fa(op)742 3544 y Fn(.)149 3660 y(3.)52 b(Let)35 b Fj(F)42 b Fn(:)32 b Fj(C)38 b(\000)-60 b(!)33 b(D)k Fn(b)s(e)f(an)f(equiv)-5 b(alence)35 b(with)g(resp)s(ect)i(to)e Fj(G)j Fn(:)32 b Fj(D)j(\000)-60 b(!)32 b(C)6 b Fn(,)37 b Fh(')32 b Fn(:)g Fj(G)6 b(F)3329 3633 y(\030)3330 3665 y Fn(=)3438 3660 y(Id)3528 3675 y Fd(C)3573 3660 y Fn(,)0 3777 y(and)32 b Fh( )f Fn(:)d Fj(F)10 b(G)512 3749 y(\030)513 3781 y Fn(=)617 3777 y(Id)707 3792 y Fd(D)768 3777 y Fn(.)43 b(Sho)m(w)32 b(that)f Fj(G)j Fn(:)28 b Fj(D)i(\000)-60 b(!)28 b(C)37 b Fn(is)31 b(an)h(equiv)-5 b(alence.)43 b(Sho)m(w)32 b(that)f Fj(G)38 b Fn(is)31 b(uniquely)0 3893 y(determined)h(b)m(y)i Fj(F)42 b Fn(up)33 b(to)f(a)g(natural)g (isomorphism.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF