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b(from)g(the)i(Y)-8 b(oneda)33 b(Lemma)e(with)h Fk(F)38 b Fp(=)27 b(Mor)2678 4630 y Fi(C)2723 4615 y Fp(\()p Fj(A;)17 b Fp(-\).)149 4731 y(2.)69 b(Observ)m(e)43 b(that)d Fj(h)952 4695 y Fh(f)998 4731 y Fp(\()p Fj(C)7 b Fp(\)\()p Fj(g)t Fp(\))41 b(=)g(Mor)1612 4746 y Fi(C)1658 4731 y Fp(\()p Fj(A;)17 b(g)t Fp(\)\()p Fj(f)11 b Fp(\))40 b(=)i Fj(g)t(f)52 b Fp(=)41 b(Mor)2639 4746 y Fi(C)2685 4731 y Fp(\()p Fj(f)5 b(;)17 b(C)7 b Fp(\)\()p Fj(g)t Fp(\))39 b(hence)k Fj(h)3437 4695 y Fh(f)3524 4731 y Fp(=)0 4848 y(Mor)176 4863 y Fi(C)221 4848 y Fp(\()p Fj(f)5 b(;)17 b Fp(-\).)39 b(Since)22 b(w)m(e)h(ha)m(v)m(e)g(Mor)1260 4863 y Fi(C)1305 4848 y Fp(\()p Fj(f)5 b(;)17 b Fp(-\)Mor)1687 4863 y Fi(C)1732 4848 y Fp(\()p Fj(g)t(;)g Fp(-)n(\))28 b(=)f(Mor)2242 4863 y Fi(C)2287 4848 y Fp(\()p Fj(g)t(f)5 b(;)17 b Fp(-)n(\))22 b(and)g(Mor)2919 4863 y Fi(C)2964 4848 y Fp(\()p Fj(f)5 b(;)17 b Fp(-)o(\))28 b(=)g(id)3382 4863 y Fg(Mor)3509 4874 y Fc(C)3549 4863 y Fg(\()p Fh(A;)p Fp(-)p Fg(\))0 4964 y Fp(if)e(and)i(only)e(if)g Fj(f)39 b Fp(=)27 b(1)800 4979 y Fh(A)884 4964 y Fp(w)m(e)i(get)e(the)h(one-to-one)e(corresp)s (ondence)k(b)s(et)m(w)m(een)f(the)f(isomorphisms)0 5080 y(from)j(1.)149 5196 y(3.)44 b(and)32 b(4.)44 b(follo)m(w)30 b(b)m(y)k(dualizing.)p 3529 5196 V 3533 5134 59 4 v 3533 5196 V 3591 5196 4 66 v eop %%Page: 41 3 41 2 bop 777 236 a Fn(9.)76 b(ADJOINT)33 b(FUNCTORS)f(AND)g(THE)i (YONED)n(A)d(LEMMA)675 b(41)149 425 y Fm(Remark)37 b(8.9.3.)50 b Fp(The)33 b(map)f Fj(\031)37 b Fp(is)32 b(a)h(natural)e (transformation)g(in)h(the)h(argumen)m(ts)g Fj(A)g Fp(and)0 541 y Fk(F)10 b Fp(.)76 b(More)44 b(precisely:)66 b(if)42 b Fj(f)57 b Fp(:)47 b Fj(A)f Fk(\000)-60 b(!)46 b Fj(B)j Fp(and)44 b Fj(\036)i Fp(:)g Fk(F)56 b(\000)-60 b(!)47 b(G)i Fp(are)44 b(giv)m(en)f(then)i(the)f(follo)m(wing)0 658 y(diagrams)31 b(comm)m(ute)1199 1252 y(Nat\(Mor)1573 1267 y Fi(C)1618 1252 y Fp(\()p Fj(A;)17 b Fp(-\))p Fj(;)g Fk(G)6 b Fp(\))277 b Fk(G)6 b Fp(\()p Fj(A)p Fp(\))p 2019 1229 221 4 v 2157 1227 a Fe(-)2108 1275 y Fh(\031)1191 765 y Fp(Nat\(Mor)1565 780 y Fi(C)1610 765 y Fp(\()p Fj(A;)17 b Fp(-)o(\))p Fj(;)g Fk(F)10 b Fp(\))260 b Fk(F)10 b Fp(\()p Fj(A)p Fp(\))p 2028 741 203 4 v 2148 739 a Fe(-)2108 721 y Fh(\031)p 1593 1159 4 351 v 1595 1159 a Fe(?)1036 1000 y Fg(Nat)q(\(Mor\()p Fh(A;)p Fp(-)o Fg(\))p Fh(;\036)p Fg(\))p 2373 1159 V 2375 1159 a Fe(?)2414 1000 y Fh(\036)p Fg(\()p Fh(A)p Fg(\))1186 1877 y Fp(Nat\(Mor)1560 1892 y Fi(C)1605 1877 y Fp(\()p Fj(B)5 b(;)17 b Fp(-)o(\))p Fj(;)g Fk(F)10 b Fp(\))241 b Fk(F)10 b Fp(\()p Fj(B)5 b Fp(\))p Fj(:)p 2029 1854 184 4 v 2130 1852 a Fe(-)2099 1900 y Fh(\031)1189 1390 y Fp(Nat\(Mor)1563 1405 y Fi(C)1608 1390 y Fp(\()p Fj(A;)17 b Fp(-)o(\))p Fj(;)g Fk(F)10 b Fp(\))261 b Fk(F)10 b Fp(\()p Fj(A)p Fp(\))p 2026 1366 203 4 v 2146 1364 a Fe(-)2106 1346 y Fh(\031)p 1591 1784 4 351 v 1593 1784 a Fe(?)1035 1625 y Fg(Nat)q(\(Mor)o(\()p Fh(f)s(;)p Fp(-)p Fg(\))p Fh(;)p Fi(F)d Fg(\))p 2371 1784 V 2373 1784 a Fe(?)2412 1625 y Fi(F)g Fg(\()p Fh(f)g Fg(\))0 2010 y Fp(This)33 b(can)g(b)s(e)f(easily)g(c)m(hec)m(k)m(ed.)47 b(F)-8 b(urthermore)32 b(w)m(e)h(ha)m(v)m(e)h(for)e Fj( )g Fp(:)c(Mor)2607 2025 y Fi(C)2652 2010 y Fp(\()p Fj(A;)17 b Fp(-)o(\))28 b Fk(\000)-60 b(!)28 b(F)135 2171 y Fj(\031)20 b Fp(Nat\(Mor)584 2186 y Fi(C)629 2171 y Fp(\()p Fj(A;)d Fp(-\))p Fj(;)g(\036)p Fp(\)\()p Fj( )t Fp(\))26 b(=)i Fj(\031)t Fp(\()p Fj(\036 )t Fp(\))f(=)h(\()p Fj(\036 )t Fp(\)\()p Fj(A)p Fp(\)\(1)2096 2186 y Fh(A)2152 2171 y Fp(\))g(=)f Fj(\036)p Fp(\()p Fj(A)p Fp(\))p Fj( )t Fp(\()p Fj(A)p Fp(\)\(1)2831 2186 y Fh(A)2888 2171 y Fp(\))g(=)h Fj(\036)p Fp(\()p Fj(A)p Fp(\))p Fj(\031)t Fp(\()p Fj( )t Fp(\))0 2333 y(and)153 2468 y Fj(\031)20 b Fp(Nat\(Mor)602 2483 y Fi(C)647 2468 y Fp(\()p Fj(f)5 b(;)17 b Fp(-\))p Fj(;)g Fk(F)10 b Fp(\)\()p Fj( )t Fp(\))26 b(=)i Fj(\031)t Fp(\()p Fj( )t Fp(Mor)1630 2483 y Fi(C)1675 2468 y Fp(\()p Fj(f)5 b(;)17 b Fp(-)o(\)\))28 b(=)g(\()p Fj( )t Fp(Mor)2331 2483 y Fi(C)2376 2468 y Fp(\()p Fj(f)5 b(;)17 b Fp(-)o(\)\)\()p Fj(B)5 b Fp(\)\(1)2861 2483 y Fh(B)2921 2468 y Fp(\))28 b(=)f Fj( )t Fp(\()p Fj(B)5 b Fp(\)\()p Fj(f)11 b Fp(\))153 2584 y(=)27 b Fj( )t Fp(\()p Fj(B)5 b Fp(\)Mor)655 2599 y Fi(C)700 2584 y Fp(\()p Fj(A;)17 b(f)11 b Fp(\)\(1)1039 2599 y Fh(A)1095 2584 y Fp(\))27 b(=)h Fk(F)10 b Fp(\()p Fj(f)h Fp(\))p Fj( )t Fp(\()p Fj(A)p Fp(\)\(1)1784 2599 y Fh(A)1840 2584 y Fp(\))27 b(=)h Fk(F)10 b Fp(\()p Fj(f)h Fp(\))p Fj(\031)t Fp(\()p Fj( )t Fp(\))p Fj(:)149 2757 y Fm(Remark)37 b(8.9.4.)50 b Fp(By)35 b(the)h(previous)f(corollary)e(the)j(represen)m (ting)g(ob)5 b(ject)35 b Fj(A)g Fp(is)g(uniquely)0 2874 y(determined)d(up)h(to)g(isomorphism)d(b)m(y)j(the)g(isomorphism)d (class)j(of)f(the)h(functor)g(Mor)3236 2889 y Fi(C)3281 2874 y Fp(\()p Fj(A;)17 b Fp(-)o(\).)149 3051 y Fm(Problem)36 b(8.9.1.)49 b Fp(1.)j(Determine)35 b(explicitly)e(all)h(natural)g (endomorphisms)h(from)f Fb(G)3430 3066 y Fh(a)3513 3051 y Fp(to)0 3167 y Fb(G)71 3182 y Fh(a)152 3167 y Fp(\(as)e(de\014ned)i (in)e(Lemma)f(2.3.5\).)149 3283 y(2.)44 b(Determine)31 b(all)g(additiv)m(e)h(natural)f(endomorphisms)h(of)g Fb(G)2472 3298 y Fh(a)2520 3283 y Fp(.)149 3399 y(3.)44 b(Determine)31 b(all)g(natural)g(transformations)g(from)g Fb(G)2226 3414 y Fh(a)2306 3399 y Fp(to)h Fb(G)2497 3414 y Fh(m)2602 3399 y Fp(\(see)h(Lemma)f(2.3.7\).)149 3516 y(4.)44 b(Determine)31 b(all)g(natural)g(automorphisms)g(of)h Fb(G)2088 3531 y Fh(m)2160 3516 y Fp(.)149 3693 y Fm(Prop)s(osition)k (8.9.5.)49 b Fl(L)-5 b(et)46 b Fk(G)52 b Fp(:)47 b Fk(C)35 b(\002)c(D)49 b(\000)-58 b(!)46 b Fm(Set)f Fl(b)-5 b(e)45 b(a)g(c)-5 b(ovariant)44 b(bifunctor)h(such)g(that)0 3809 y(the)c(functor)g Fk(G)6 b Fp(\()p Fj(C)r(;)17 b Fl(-)p Fp(\))39 b(:)g Fk(D)j(\000)-58 b(!)40 b Fm(Set)h Fl(is)g(r)-5 b(epr)g(esentable)40 b(for)h(al)5 b(l)40 b Fj(C)47 b Fk(2)39 b(C)6 b Fl(.)64 b(Then)40 b(ther)-5 b(e)41 b(exists)g(a)0 3925 y(c)-5 b(ontr)g(avariant)37 b(functor)g Fk(F)42 b Fp(:)33 b Fk(C)38 b(\000)-58 b(!)33 b(D)40 b Fl(such)d(that)h Fk(G)2002 3898 y(\030)2002 3929 y Fp(=)2111 3925 y(Mor)2288 3940 y Fi(D)2348 3925 y Fp(\()p Fk(F)10 b Fl(-)p Fj(;)17 b Fl(-)o Fp(\))37 b Fl(holds.)52 b(F)-7 b(urthermor)i(e)37 b Fk(F)0 4041 y Fl(is)e(uniquely)g(determine)-5 b(d)34 b(by)h Fk(G)41 b Fl(up)35 b(to)g(isomorphism.)149 4218 y Ff(Pr)n(oof.)42 b Fp(F)-8 b(or)39 b(eac)m(h)h Fj(C)46 b Fk(2)39 b(C)45 b Fp(c)m(ho)s(ose)40 b(an)f(ob)5 b(ject)40 b Fk(F)10 b Fp(\()p Fj(C)d Fp(\))38 b Fk(2)i(D)h Fp(and)f(an)f(isomorphism)d Fj(\030)3475 4233 y Fh(C)3573 4218 y Fp(:)0 4335 y Fk(G)6 b Fp(\()p Fj(C)r(;)17 b Fp(-)o(\))326 4307 y Fk(\030)327 4339 y Fp(=)441 4335 y(Mor)617 4350 y Fi(D)678 4335 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)d Fp(\))p Fj(;)17 b Fp(-)n(\).)61 b(Giv)m(en)38 b Fj(f)48 b Fp(:)38 b Fj(C)44 b Fk(\000)-60 b(!)37 b Fj(C)1946 4298 y Fi(0)2008 4335 y Fp(in)g Fk(C)45 b Fp(then)39 b(let)e Fk(F)10 b Fp(\()p Fj(f)h Fp(\))37 b(:)g Fk(F)10 b Fp(\()p Fj(C)3113 4298 y Fi(0)3136 4335 y Fp(\))37 b Fk(\000)-60 b(!)38 b(F)10 b Fp(\()p Fj(C)d Fp(\))0 4451 y(b)s(e)38 b(the)h(uniquely)f(determined)g(morphism)e(\(b) m(y)j(the)f(Y)-8 b(oneda)39 b(Lemma\))d(in)i Fk(D)i Fp(suc)m(h)g(that)e (the)0 4567 y(diagram)1152 5190 y Fk(G)6 b Fp(\()p Fj(C)1332 5154 y Fi(0)1356 5190 y Fj(;)17 b Fp(-)o(\))298 b(Mor)1944 5205 y Fi(D)2005 5190 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)2240 5154 y Fi(0)2262 5190 y Fp(\))p Fj(;)17 b Fp(-)o(\))p 1498 5167 241 4 v 1656 5165 a Fe(-)1566 5230 y Fh(\030)1597 5249 y Fd(C)1644 5235 y Fc(0)1167 4702 y Fk(G)6 b Fp(\()p Fj(C)r(;)17 b Fp(-)o(\))323 b(Mor)1956 4717 y Fi(D)2016 4702 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)d Fp(\))p Fj(;)17 b Fp(-)o(\))p 1485 4679 267 4 v 1669 4677 a Fe(-)1576 4645 y Fh(\030)1607 4656 y Fd(C)p 1309 5097 4 351 v 1311 5097 a Fe(?)1082 4938 y Fi(G)t Fg(\()p Fh(f)s(;)p Fp(-)p Fg(\))p 2089 5097 V 2091 5097 a Fe(?)2130 4938 y Fg(Mor)o(\()p Fi(F)7 b Fg(\()p Fh(f)g Fg(\))p Fh(;)p Fp(-)q Fg(\))p eop %%Page: 42 4 42 3 bop 0 236 a Fn(42)1456 b(8.)77 b(TOOLBO)n(X)0 425 y Fp(comm)m(utes.)f(Because)44 b(of)f(the)h(uniqueness)h Fk(F)10 b Fp(\()p Fj(f)h Fp(\))42 b(and)i(b)s(ecause)g(of)f(the)h (functorialit)m(y)d(of)i Fk(G)0 541 y Fp(it)38 b(is)g(easy)i(to)f(see)h (that)e Fk(F)10 b Fp(\()p Fj(f)h(g)t Fp(\))37 b(=)h Fk(F)10 b Fp(\()p Fj(g)t Fp(\))p Fk(F)g Fp(\()p Fj(f)h Fp(\))37 b(and)i Fk(F)10 b Fp(\(1)2183 556 y Fh(C)2241 541 y Fp(\))38 b(=)g(1)2480 557 y Fi(F)7 b Fg(\()p Fh(C)e Fg(\))2690 541 y Fp(hold)38 b(and)h(that)g Fk(F)48 b Fp(is)38 b(a)0 658 y(con)m(tra)m(v)-5 b(arian)m(t)32 b(functor.)149 774 y(If)24 b Fk(F)320 738 y Fi(0)371 774 y Fp(:)k Fk(C)34 b(\000)-60 b(!)27 b(D)g Fp(is)c(giv)m(en)h(with)g Fk(G)1402 746 y(\030)1403 778 y Fp(=)1507 774 y(Mor)1683 789 y Fi(D)1744 774 y Fp(\()p Fk(F)1864 738 y Fi(0)1887 774 y Fp(-)o Fj(;)17 b Fp(-)o(\))24 b(then)h Fj(\036)i Fp(:)h(Mor)2587 789 y Fi(D)2648 774 y Fp(\()p Fk(F)10 b Fp(-)o Fj(;)17 b Fp(-)o(\))2942 746 y Fk(\030)2942 778 y Fp(=)3047 774 y(Mor)3223 789 y Fi(D)3284 774 y Fp(\()p Fk(F)3404 738 y Fi(0)3426 774 y Fp(-)p Fj(;)g Fp(-)o(\).)0 890 y(Hence)32 b(b)m(y)g(the)f(Y)-8 b(oneda)31 b(Lemma)e Fj( )t Fp(\()p Fj(C)7 b Fp(\))28 b(:)f Fk(F)10 b Fp(\()p Fj(C)d Fp(\))1843 862 y Fk(\030)1844 894 y Fp(=)1949 890 y Fk(F)2031 854 y Fi(0)2053 890 y Fp(\()p Fj(C)g Fp(\))31 b(is)f(an)h(isomorphism)d (for)i(all)f Fj(C)35 b Fk(2)28 b(C)6 b Fp(.)0 1006 y(With)32 b(these)i(isomorphisms)c(induced)k(b)m(y)f Fj(\036)f Fp(the)h(diagram)696 1732 y(Mor)872 1747 y Fi(D)933 1732 y Fp(\()p Fk(F)1053 1696 y Fi(0)1076 1732 y Fp(\()p Fj(C)1191 1696 y Fi(0)1214 1732 y Fp(\))p Fj(;)17 b Fp(-)o(\))902 b(Mor)2444 1747 y Fi(D)2505 1732 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)2740 1696 y Fi(0)2762 1732 y Fp(\))p Fj(;)17 b Fp(-)o(\))p 1395 1709 845 4 v 2157 1707 a Fe(-)1609 1777 y Fg(Mor\()p Fh( )r Fg(\()p Fh(C)1894 1754 y Fc(0)1918 1777 y Fg(\))p Fh(;)p Fp(-)o Fg(\))708 1245 y Fp(Mor)884 1260 y Fi(D)945 1245 y Fp(\()p Fk(F)1065 1208 y Fi(0)1088 1245 y Fp(\()p Fj(C)7 b Fp(\))p Fj(;)17 b Fp(-)o(\))925 b(Mor)2456 1260 y Fi(D)2516 1245 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)d Fp(\))p Fj(;)17 b Fp(-)o(\))p 1383 1221 868 4 v 2168 1219 a Fe(-)1621 1184 y Fg(Mor)o(\()p Fh( )r Fg(\()p Fh(C)5 b Fg(\))p Fh(;)p Fp(-)q Fg(\))p 1030 1639 4 351 v 1031 1639 a Fe(?)582 1481 y Fg(Mor\()p Fi(F)794 1457 y Fc(0)817 1481 y Fg(\()p Fh(f)i Fg(\))p Fh(;)p Fp(-)p Fg(\))p 2590 1639 V 2591 1639 a Fe(?)2630 1480 y Fg(Mor\()p Fi(F)g Fg(\()p Fh(f)g Fg(\))p Fh(;)p Fp(-)p Fg(\))0 1894 y Fp(comm)m(utes.)43 b(Hence)34 b(the)f(diagram)1402 2563 y Fk(F)10 b Fp(\()p Fj(C)d Fp(\))338 b Fk(F)2057 2527 y Fi(0)2080 2563 y Fp(\()p Fj(C)7 b Fp(\))p 1665 2540 281 4 v 1863 2538 a Fe(-)1726 2607 y Fh( )r Fg(\()p Fh(C)e Fg(\))1390 2076 y Fk(F)10 b Fp(\()p Fj(C)1587 2039 y Fi(0)1610 2076 y Fp(\))315 b Fk(F)2045 2039 y Fi(0)2068 2076 y Fp(\()p Fj(C)2183 2039 y Fi(0)2206 2076 y Fp(\))p 1677 2052 258 4 v 1852 2050 a Fe(-)1715 2015 y Fh( )r Fg(\()p Fh(C)1845 1992 y Fc(0)1868 2015 y Fg(\))p 1517 2470 4 351 v 1519 2470 a Fe(?)1304 2312 y Fi(F)1361 2288 y Fc(0)1384 2312 y Fg(\()p Fh(f)7 b Fg(\))p 2102 2470 V 2104 2470 a Fe(?)2143 2311 y Fi(F)g Fg(\()p Fh(f)g Fg(\))0 2749 y Fp(comm)m(utes.)43 b(Th)m(us)35 b Fj( )c Fp(:)d Fk(F)37 b(\000)-60 b(!)28 b(F)1229 2713 y Fi(0)1284 2749 y Fp(is)k(a)h(natural)e(isomorphism.)p 3529 2749 4 66 v 3533 2686 59 4 v 3533 2749 V 3591 2749 4 66 v 149 2925 a Fm(De\014nition)37 b(8.9.6.)49 b Fp(Let)38 b Fk(C)44 b Fp(and)38 b Fk(D)j Fp(b)s(e)d(categories)g(and)g Fk(F)46 b Fp(:)37 b Fk(C)44 b(\000)-60 b(!)37 b(D)j Fp(and)e Fk(G)43 b Fp(:)37 b Fk(D)j(\000)-60 b(!)37 b(C)0 3041 y Fp(b)s(e)d(co)m(v)-5 b(arian)m(t)32 b(functors.)47 b Fk(F)c Fp(is)32 b(called)h Fl(leftadjoint)f Fp(to)h Fk(G)40 b Fp(and)33 b Fk(G)40 b Fl(rightadjoint)32 b Fp(to)h Fk(F)43 b Fp(if)32 b(there)i(is)0 3157 y(a)g(natural)g (isomorphism)e(of)j(bifunctors)f Fj(\036)e Fp(:)f(Mor)1897 3172 y Fi(D)1958 3157 y Fp(\()p Fk(F)10 b Fp(-)o Fj(;)17 b Fp(-)o(\))31 b Fk(\000)-60 b(!)32 b Fp(Mor)2580 3172 y Fi(C)2625 3157 y Fp(\(-)p Fj(;)17 b Fk(G)6 b Fp(-)o(\))35 b(from)e Fk(C)3200 3121 y Fh(op)3298 3157 y Fk(\002)24 b(D)37 b Fp(to)0 3274 y Fm(Set)p Fp(.)149 3450 y Fm(Lemma)h(8.9.7.)49 b Fl(If)41 b Fk(F)50 b Fp(:)40 b Fk(C)46 b(\000)-58 b(!)41 b(D)j Fl(is)d(leftadjoint)g(to)h Fk(G)k Fp(:)41 b Fk(D)h(\000)-58 b(!)41 b(C)48 b Fl(then)41 b Fk(F)51 b Fl(is)42 b(uniquely)0 3566 y(determine)-5 b(d)36 b(by)h Fk(G)43 b Fl(up)37 b(to)g(isomorphism.)48 b(Similarly)36 b Fk(G)43 b Fl(is)37 b(uniquely)g(determine)-5 b(d)36 b(by)h Fk(F)46 b Fl(up)37 b(to)0 3682 y(isomorphism.)149 3858 y Ff(Pr)n(oof.)42 b Fp(No)m(w)f(w)m(e)g(pro)m(v)m(e)h(the)e(\014rst)h(claim.)63 b(Assume)41 b(that)f(also)f Fk(F)2763 3822 y Fi(0)2826 3858 y Fp(is)h(leftadjoin)m(t)e(to)i Fk(G)0 3975 y Fp(with)c Fj(\036)284 3939 y Fi(0)341 3975 y Fp(:)e(Mor)579 3990 y Fi(D)639 3975 y Fp(\()p Fk(F)759 3939 y Fi(0)782 3975 y Fp(-)o Fj(;)17 b Fp(-\))34 b Fk(\000)-60 b(!)34 b Fp(Mor)1290 3990 y Fi(C)1335 3975 y Fp(\(-)p Fj(;)17 b Fk(G)6 b Fp(-)o(\).)55 b(Then)37 b(w)m(e)h(ha)m(v)m(e)f(a)g(natural)e(isomorphism)f Fj(\036)3364 3939 y Fi(0)3387 3929 y(\000)p Fg(1)3481 3975 y Fj(\036)g Fp(:)0 4092 y(Mor)176 4107 y Fi(D)237 4092 y Fp(\()p Fk(F)10 b Fp(-)o Fj(;)17 b Fp(-)o(\))28 b Fk(\000)-60 b(!)27 b Fp(Mor)851 4107 y Fi(D)912 4092 y Fp(\()p Fk(F)1032 4056 y Fi(0)1055 4092 y Fp(-)o Fj(;)17 b Fp(-\).)43 b(By)33 b(Prop)s(osition)e(8.9.5)h(w)m(e)i(get)e Fk(F)2598 4064 y(\030)2599 4096 y Fp(=)2703 4092 y Fk(F)2785 4056 y Fi(0)2808 4092 y Fp(.)p 3529 4092 V 3533 4029 59 4 v 3533 4092 V 3591 4092 4 66 v 149 4268 a Fm(Lemma)38 b(8.9.8.)49 b Fl(A)43 b(functor)g Fk(G)48 b Fp(:)43 b Fk(D)i(\000)-58 b(!)42 b(C)49 b Fl(has)43 b(a)f(leftadjoint)g(functor)h (i\013)g(al)5 b(l)42 b(functors)0 4384 y Fp(Mor)176 4399 y Fi(C)221 4384 y Fp(\()p Fj(C)r(;)17 b Fk(G)6 b Fl(-)o Fp(\))35 b Fl(ar)-5 b(e)35 b(r)-5 b(epr)g(esentable.)149 4560 y Ff(Pr)n(oof.)42 b Fp(follo)m(ws)32 b(from)f(8.9.5.)p 3529 4560 V 3533 4497 59 4 v 3533 4560 V 3591 4560 4 66 v 149 4736 a Fm(Lemma)38 b(8.9.9.)49 b Fl(L)-5 b(et)35 b Fk(F)i Fp(:)28 b Fk(C)34 b(\000)-58 b(!)28 b(D)38 b Fl(and)c Fk(G)g Fp(:)28 b Fk(D)i(\000)-58 b(!)28 b(C)41 b Fl(b)-5 b(e)35 b(c)-5 b(ovariant)34 b(functors.)45 b(Then)551 4896 y Fp(Nat\(Id)838 4911 y Fi(C)883 4896 y Fj(;)17 b Fk(G)6 b(F)k Fp(\))27 b Fk(3)i Fp(\010)f Fk(7!)f(G)6 b Fl(-)p Fp(\010)p Fl(-)28 b Fk(2)g Fp(Nat\(Mor)2160 4911 y Fi(D)2221 4896 y Fp(\()p Fk(F)10 b Fl(-)o Fj(;)17 b Fl(-)p Fp(\))p Fj(;)g Fp(Mor)2711 4911 y Fi(C)2756 4896 y Fp(\()p Fl(-)p Fj(;)g Fk(G)6 b Fl(-)p Fp(\)\))0 5056 y Fl(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map) 372 5216 y Fp(Nat\(Mor)746 5231 y Fi(D)807 5216 y Fp(\()p Fk(F)10 b Fl(-)o Fj(;)17 b Fl(-)p Fp(\))p Fj(;)g Fp(Mor)1297 5231 y Fi(C)1342 5216 y Fp(\()p Fl(-)p Fj(;)g Fk(G)6 b Fl(-)p Fp(\)\))27 b Fk(3)i Fj(\036)e Fk(7!)g Fj(\036)p Fp(\()p Fl(-)p Fj(;)17 b Fk(F)10 b Fl(-)o Fp(\)\(1)2385 5231 y Fi(F)d Fl(-)2481 5216 y Fp(\))27 b Fk(2)h Fp(Nat\(Id)2927 5231 y Fi(C)2972 5216 y Fj(;)17 b Fk(G)6 b(F)k Fp(\))p Fj(:)p eop %%Page: 43 5 43 4 bop 777 236 a Fn(9.)76 b(ADJOINT)33 b(FUNCTORS)f(AND)g(THE)i (YONED)n(A)d(LEMMA)675 b(43)0 425 y Fl(F)-7 b(urthermor)i(e)538 607 y Fp(Nat)o(\()p Fk(F)10 b(G)c Fj(;)17 b Fp(Id)1015 622 y Fi(C)1060 607 y Fp(\))28 b Fk(3)g Fp(\011)g Fk(7!)f Fp(\011)p Fl(-)p Fk(F)10 b Fl(-)27 b Fk(2)h Fp(Nat\(Mor)2174 622 y Fi(C)2219 607 y Fp(\()p Fl(-)o Fj(;)17 b Fk(G)6 b Fl(-)p Fp(\))p Fj(;)17 b Fp(Mor)2693 622 y Fi(D)2754 607 y Fp(\()p Fk(F)10 b Fl(-)o Fj(;)17 b Fl(-)o Fp(\)\))0 789 y Fl(is)35 b(a)f(bije)-5 b(ctive)34 b(map)h(with)f(inverse)g(map) 377 971 y Fp(Nat)o(\(Mor)751 986 y Fi(C)796 971 y Fp(\()p Fl(-)o Fj(;)17 b Fk(G)6 b Fl(-)p Fp(\))p Fj(;)17 b Fp(Mor)1270 986 y Fi(D)1331 971 y Fp(\()p Fk(F)10 b Fl(-)o Fj(;)17 b Fl(-)o Fp(\)\))28 b Fk(3)g Fj( )k Fk(7!)27 b Fj( )t Fp(\()p Fk(G)6 b Fl(-)p Fj(;)17 b Fl(-)p Fp(\)\(1)2392 986 y Fi(G)t Fl(-)2476 971 y Fp(\))28 b Fk(2)g Fp(Nat\()p Fk(F)10 b(G)c Fj(;)17 b Fp(Id)3113 986 y Fi(C)3158 971 y Fp(\))p Fj(:)149 1160 y Ff(Pr)n(oof.)42 b Fp(The)i(natural)e (transformation)e Fk(G)6 b Fp(-\010-)43 b(is)f(de\014ned)i(as)f(follo)m (ws.)73 b(Giv)m(en)43 b Fj(C)52 b Fk(2)46 b(C)6 b Fp(,)0 1276 y Fj(D)53 b Fk(2)e(D)d Fp(and)e Fj(f)61 b Fk(2)51 b Fp(Mor)981 1291 y Fi(D)1042 1276 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)d Fp(\))p Fj(;)17 b(D)s Fp(\))44 b(then)j(let)e(\()p Fk(G)6 b Fp(-\010-\)\()p Fj(C)r(;)17 b(D)s Fp(\)\()p Fj(f)11 b Fp(\))48 b(:=)j Fk(G)6 b Fp(\()p Fj(f)11 b Fp(\)\010\()p Fj(C)c Fp(\))50 b(:)h Fj(C)57 b Fk(\000)-60 b(!)0 1392 y(G)6 b(F)k Fp(\()p Fj(C)d Fp(\))27 b Fk(\000)-60 b(!)28 b(G)6 b Fp(\()p Fj(D)s Fp(\).)43 b(It)33 b(is)f(easy)h(to)g(c)m (hec)m(k)i(the)e(prop)s(erties)f(of)g(a)g(natural)g(transformation.)149 1509 y(Giv)m(en)51 b(\010)h(then)g(one)f(obtains)f(b)m(y)i(comp)s (osition)d(of)i(the)g(t)m(w)m(o)h(maps)f Fk(G)6 b Fp(\(1)3033 1524 y Fi(F)h Fg(\()p Fh(C)e Fg(\))3204 1509 y Fp(\)\010\()p Fj(C)i Fp(\))59 b(=)0 1625 y Fk(G)6 b(F)k Fp(\(1)234 1640 y Fh(C)293 1625 y Fp(\)\010\()p Fj(C)d Fp(\))27 b(=)h(\010\()p Fj(C)7 b Fp(\).)44 b(Giv)m(en)32 b Fj(\036)g Fp(one)h(obtains)531 1805 y Fk(G)6 b Fp(\()p Fj(f)11 b Fp(\)\()p Fj(\036)p Fp(\()p Fj(C)r(;)17 b Fk(F)10 b Fp(\()p Fj(C)d Fp(\)\)\(1)1341 1821 y Fi(F)g Fg(\()p Fh(C)e Fg(\))1509 1805 y Fp(\))28 b(=)f(Mor)1855 1820 y Fi(C)1900 1805 y Fp(\()p Fj(C)r(;)17 b Fk(G)6 b Fp(\()p Fj(f)11 b Fp(\)\))p Fj(\036)p Fp(\()p Fj(C)r(;)17 b Fk(F)10 b Fp(\()p Fj(C)d Fp(\)\)\(1)2864 1821 y Fi(F)g Fg(\()p Fh(C)e Fg(\))3031 1805 y Fp(\))531 1921 y(=)27 b Fj(\036)p Fp(\()p Fj(C)r(;)17 b(D)s Fp(\)Mor)1143 1936 y Fi(D)1204 1921 y Fp(\()p Fk(F)10 b Fp(\()p Fj(C)d Fp(\))p Fj(;)17 b(f)11 b Fp(\)\(1)1705 1937 y Fi(F)c Fg(\()p Fh(C)e Fg(\))1874 1921 y Fp(\))28 b(=)f Fj(\036)p Fp(\()p Fj(C)r(;)17 b(D)s Fp(\)\()p Fj(f)11 b Fp(\))p Fj(:)0 2100 y Fp(The)34 b(second)f(part)g (of)f(the)h(lemma)d(is)i(pro)m(v)m(ed)i(similarly)-8 b(.)p 3529 2100 4 66 v 3533 2037 59 4 v 3533 2100 V 3591 2100 4 66 v 149 2288 a Fm(Prop)s(osition)36 b(8.9.10.)49 b Fl(L)-5 b(et)285 2470 y Fj(\036)28 b Fp(:)g(Mor)602 2485 y Fi(D)662 2470 y Fp(\()p Fk(F)10 b Fl(-)p Fj(;)17 b Fl(-)o Fp(\))28 b Fk(\000)-58 b(!)28 b Fp(Mor)1284 2485 y Fi(C)1329 2470 y Fp(\()p Fl(-)p Fj(;)17 b Fk(G)6 b Fl(-)p Fp(\))134 b Fl(and)g Fj( )32 b Fp(:)c(Mor)2333 2485 y Fi(C)2378 2470 y Fp(\()p Fl(-)o Fj(;)17 b Fk(G)6 b Fl(-)p Fp(\))28 b Fk(\000)-58 b(!)28 b Fp(Mor)2983 2485 y Fi(D)3044 2470 y Fp(\()p Fk(F)10 b Fl(-)o Fj(;)17 b Fl(-)p Fp(\))0 2652 y Fl(b)-5 b(e)31 b(natur)-5 b(al)32 b(tr)-5 b(ansformations)31 b(with)h(asso)-5 b(ciate)g(d)30 b(natur)-5 b(al)32 b(tr)-5 b(ansformations)31 b(\(by)h(L)-5 b(emma)31 b(8.9.9\))0 2768 y Fp(\010)d(:)g(Id)242 2783 y Fi(C)315 2768 y Fk(\000)-58 b(!)28 b(G)6 b(F)45 b Fl(r)-5 b(esp.)44 b Fp(\011)28 b(:)f Fk(F)10 b(G)34 b(\000)-58 b(!)28 b Fp(Id)1458 2783 y Fi(D)1519 2768 y Fl(.)149 2908 y(1\))35 b(Then)f(we)g(have)h Fj(\036 )c Fp(=)d(id)1234 2923 y Fg(Mor)o(\()p Fl(-)q Fh(;)p Fi(G)t Fl(-)o Fg(\))1590 2908 y Fl(if)35 b(and)f(only)h(if)g Fp(\()p Fk(G)2343 2851 y Fg(\010)p Fi(G)2311 2908 y Fk(\000)-16 b(!)28 b(G)6 b(F)k(G)2769 2851 y Fi(G)t Fg(\011)2739 2908 y Fk(\000)-16 b(!)27 b(G)6 b Fp(\))28 b(=)g(id)3243 2923 y Fi(G)3293 2908 y Fl(.)149 3060 y(2\))35 b(We)g(also)f(have)g Fj( )t(\036)28 b Fp(=)f(id)1212 3076 y Fg(Mor)o(\()p Fi(F)7 b Fl(-)p Fh(;)p Fl(-)p Fg(\))1579 3060 y Fl(if)35 b(and)f(only)h(if)g Fp(\()p Fk(F)2343 3004 y Fi(F)7 b Fg(\010)2317 3060 y Fk(\000)-16 b(!)27 b(F)10 b(G)c(F)2785 3004 y Fg(\011)p Fi(F)2761 3060 y Fk(\000)-16 b(!)27 b(F)10 b Fp(\))27 b(=)h(id)3281 3075 y Fi(F)3342 3060 y Fl(.)149 3249 y Ff(Pr)n(oof.)42 b Fp(W)-8 b(e)33 b(get)759 3424 y Fk(G)6 b Fp(\011\()p Fj(D)s Fp(\)\010)p Fk(G)g Fp(\()p Fj(D)s Fp(\))27 b(=)h Fk(G)6 b Fp(\011\()p Fj(D)s Fp(\))p Fj(\036)p Fp(\()p Fk(G)g Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b Fk(F)10 b(G)c Fp(\()p Fj(D)s Fp(\)\)\(1)2584 3439 y Fi(F)h(G)t Fg(\()p Fh(D)r Fg(\))2803 3424 y Fp(\))759 3540 y(=)27 b(Mor)1039 3555 y Fi(C)1084 3540 y Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b Fk(G)6 b Fp(\011\()p Fj(D)s Fp(\)\))p Fj(\036)p Fp(\()p Fk(G)g Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b Fk(F)10 b(G)c Fp(\()p Fj(D)s Fp(\)\)\(1)2527 3555 y Fi(F)h(G)t Fg(\()p Fh(D)r Fg(\))2745 3540 y Fp(\))759 3656 y(=)27 b Fj(\036)p Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b(D)s Fp(\)Mor)1524 3671 y Fi(D)1585 3656 y Fp(\()p Fk(F)10 b(G)c Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b Fp(\011\()p Fj(D)s Fp(\)\)\(1)2335 3672 y Fi(F)7 b(G)t Fg(\()p Fh(D)r Fg(\))2555 3656 y Fp(\))759 3772 y(=)27 b Fj(\036)p Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b(D)s Fp(\)\(\011\()p Fj(D)s Fp(\)\))759 3889 y(=)27 b Fj(\036)p Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b(D)s Fp(\))p Fj( )t Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b(D)s Fp(\)\(1)1932 3904 y Fi(G)t Fg(\()p Fh(D)r Fg(\))2095 3889 y Fp(\))759 4005 y(=)27 b Fj(\036 )t Fp(\()p Fk(G)6 b Fp(\()p Fj(D)s Fp(\))p Fj(;)17 b(D)s Fp(\)\(1)1503 4020 y Fi(G)t Fg(\()p Fh(D)r Fg(\))1667 4005 y Fp(\))p Fj(:)0 4183 y Fp(Similarly)29 b(w)m(e)k(get)383 4358 y Fj(\036 )t Fp(\()p Fj(C)r(;)17 b(D)s Fp(\)\()p Fj(f)11 b Fp(\))26 b(=)h Fj(\036)p Fp(\()p Fj(C)r(;)17 b(D)s Fp(\))p Fj( )t Fp(\()p Fj(C)r(;)g(D)s Fp(\)\()p Fj(f)11 b Fp(\))25 b(=)i Fk(G)6 b Fp(\(\011\()p Fj(D)s Fp(\))p Fk(F)k Fp(\()p Fj(f)h Fp(\)\)\010\()p Fj(C)c Fp(\))383 4474 y(=)27 b Fk(G)6 b Fp(\011\()p Fj(D)s Fp(\))p Fk(G)g(F)k Fp(\()p Fj(f)h Fp(\)\010\()p Fj(C)c Fp(\))27 b(=)h Fk(G)6 b Fp(\011\()p Fj(D)s Fp(\)\010)p Fk(G)g Fp(\()p Fj(D)s Fp(\))p Fj(f)f(:)p 3147 4474 V 3151 4411 59 4 v 3151 4474 V 3209 4474 4 66 v 149 4659 a Fm(Corollary)36 b(8.9.11.)50 b Fl(L)-5 b(et)36 b Fk(F)k Fp(:)30 b Fk(C)37 b(\000)-58 b(!)30 b(D)39 b Fl(and)d Fk(G)g Fp(:)30 b Fk(D)j(\000)-58 b(!)31 b(C)42 b Fl(b)-5 b(e)36 b(functors.)49 b Fk(F)d Fl(is)36 b(leftadjoint)0 4775 y(to)25 b Fk(G)31 b Fl(if)25 b(and)f(only)h(if)g(ther)-5 b(e)25 b(ar)-5 b(e)25 b(natur)-5 b(al)25 b(tr)-5 b(ansformations)24 b Fp(\010)k(:)g(Id)2399 4790 y Fi(C)2472 4775 y Fk(\000)-58 b(!)28 b(G)6 b(F)35 b Fl(and)24 b Fp(\011)k(:)f Fk(F)10 b(G)34 b(\000)-58 b(!)28 b Fp(Id)3539 4790 y Fi(D)0 4891 y Fl(such)35 b(that)g Fp(\()p Fk(G)6 b Fp(\011\)\(\010)p Fk(G)g Fp(\))28 b(=)g(id)1062 4906 y Fi(G)1147 4891 y Fl(and)34 b Fp(\(\011)p Fk(F)10 b Fp(\)\()p Fk(F)g Fp(\010\))27 b(=)g(id)2009 4906 y Fi(F)2071 4891 y Fl(.)149 5080 y Fm(De\014nition)37 b(8.9.12.)49 b Fp(The)36 b(natural)d (transformations)g(\010)f(:)g(Id)2540 5095 y Fi(C)2617 5080 y Fk(\000)-60 b(!)31 b(G)6 b(F)45 b Fp(and)35 b(\011)c(:)h Fk(F)10 b(G)37 b(\000)-60 b(!)0 5196 y Fp(Id)89 5211 y Fi(D)181 5196 y Fp(giv)m(en)32 b(in)e(8.9.11)h(are)g(called)f Fl(unit)i Fp(and)f Fl(c)-5 b(ounit)31 b Fp(resp.)44 b(for)31 b(the)h(adjoin)m(t)e(functors)i Fk(F)40 b Fp(and)32 b Fk(G)6 b Fp(.)p eop %%Page: 44 6 44 5 bop 0 236 a Fn(44)1456 b(8.)77 b(TOOLBO)n(X)149 425 y Fm(Problem)36 b(8.9.2.)49 b Fp(1.)68 b(Let)1240 440 y Fh(R)1298 425 y Fj(M)1392 440 y Fh(S)1484 425 y Fp(b)s(e)41 b(a)f(bimo)s(dule.)66 b(Sho)m(w)42 b(that)e(the)h(functor)g Fj(M)e 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1239 y(d\))39 b(Sho)m(w)h(that)f(there)g(is)g(a)f(natural)g (isomorphism)e Fb(K)18 b Fp(-)5 b Fm(Alg)o Fp(\()p Fj(U)10 b Fp(\()p Fa(g)p Fp(\))p Fj(;)17 b(A)p Fp(\))2849 1211 y Fk(\030)2850 1243 y Fp(=)2965 1239 y(Lie-Alg)n(\()p Fa(g)p Fj(;)g(A)3483 1203 y Fh(L)3535 1239 y Fp(\))p Fj(:)0 1355 y Fp(Determine)32 b(the)h(corresp)s(onding)f(leftadjoin)m (t)f(functor)i(and)f(the)h(asso)s(ciated)g(unit)f(and)g(counit.)149 1539 y Fm(De\014nition)37 b(8.9.13.)49 b Fp(Let)39 b Fk(G)45 b Fp(:)38 b Fk(D)j(\000)-60 b(!)39 b(C)45 b Fp(b)s(e)39 b(a)g(co)m(v)-5 b(arian)m(t)38 b(functor.)63 b Fk(G)45 b Fp(generates)40 b(a)f Fl(\(c)-5 b(o-)0 1656 y(\)universal)34 b(pr)-5 b(oblem)32 b Fp(a)g(follo)m(ws:)149 1772 y(Giv)m(en)38 b Fj(C)43 b Fk(2)37 b(C)6 b Fp(.)58 b(Find)37 b(an)g(ob)5 b(ject)39 b Fk(F)10 b Fp(\()p Fj(C)d Fp(\))35 b Fk(2)i(D)j Fp(and)e(a)f(morphism)f Fj(\023)g Fp(:)h Fj(C)43 b Fk(\000)-60 b(!)36 b(G)6 b Fp(\()p Fk(F)k Fp(\()p Fj(C)d Fp(\)\))37 b(in)0 1888 y Fk(C)k Fp(suc)m(h)36 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(ossess)i(a)e(left-\(righ)m(t-\)adjoin)m(t)c(functor.)76 b(One)43 b(of)0 5028 y(the)33 b(essen)m(tial)g(prop)s(erties)f(for)h (this)f(is)g(that)h Fk(G)39 b Fp(preserv)m(es)c(limits)30 b(\(hence)k(direct)e(pro)s(ducts)i(and)0 5145 y(di\013erence)f(k)m (ernels\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF