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y(diagramsc)m(heme) 32 b Fm(D)j Fr(if)c Fl(lim)882 1052 y Fm( )-42 b(\000)1018 1006 y Fl(\()p Fm(G)6 b(F)k Fl(\))1268 979 y Fm(\030)1269 1011 y Fl(=)1373 1006 y Fm(G)c Fl(\(lim)1476 1052 y Fm( )-42 b(\000)1612 1006 y Fl(\()p Fm(F)10 b Fl(\)\))31 b Fr(for)h(eac)m(h)i (diagram)c Fm(F)37 b Fl(:)28 b Fm(D)i(\000)-59 b(!)27 b(C)6 b Fr(.)149 1181 y Fo(Prop)s(osition)36 b(8.10.12.)49 b Fn(Covariant)g(r)-5 b(epr)g(esentable)48 b(functors)h(pr)-5 b(eserve)49 b(limits.)88 b(Con-)0 1297 y(tr)-5 b(avariant)35 b(r)-5 b(epr)g(esentable)33 b(functors)i(map)f(c)-5 b(olimits)34 b(into)h(limits.)149 1471 y Fa(Pr)m(oof.)42 b Fr(W)-8 b(e)41 b(only)f(pro)m(v)m(e)i(the)g(\034rst)f(assertion.)68 b(The)42 b(second)g(assertion)f(is)f(dual)g(to)h(the)0 1588 y(\034rst)33 b(one.)44 b(F)-8 b(or)31 b(a)i(diagram)d Fm(F)37 b Fl(:)28 b Fm(D)i(\000)-59 b(!)27 b Fo(Set)32 b Fr(the)h(set)344 1744 y Fm(f)p Fl(\()p Fk(x)487 1759 y Fj(D)551 1744 y Fm(j)p Fk(D)d Fm(2)e Fl(Ob)17 b Fm(D)s Fk(;)g(x)1110 1759 y Fj(D)1202 1744 y Fm(2)28 b(F)10 b Fl(\()p Fk(D)s Fl(\)\))p Fm(j8)p Fl(\()p Fk(f)37 b Fl(:)28 b Fk(D)i Fm(\000)-59 b(!)27 b Fk(D)2177 1703 y Fg(0)2200 1744 y Fl(\))g Fm(2)i(D)h Fl(:)e Fm(F)10 b Fl(\()p Fk(f)h Fl(\)\()p Fk(x)2832 1759 y Fj(D)2895 1744 y Fl(\))28 b(=)f Fk(x)3119 1759 y Fj(D)3179 1740 y Ff(0)3206 1744 y Fm(g)0 1901 y Fr(is)f(a)g(limit)d(of)j Fm(F)36 b Fr(b)m(y)27 b(Problem)e(8.1.)41 b(No)m(w)27 b(let)f(a)g(diagram)e Fm(F)37 b Fl(:)28 b Fm(D)i(\000)-59 b(!)27 b(C)33 b Fr(b)s(e)26 b(giv)m(en)h(and)f(let)g Fl(lim)3307 1947 y Fm( )-42 b(\000)3443 1901 y Fl(\()p Fm(F)10 b Fl(\))0 2021 y Fr(b)s(e)35 b(the)g(limit.)46 b(F)-8 b(urthermore)33 b(let)h Fl(Mor)1467 2036 y Fg(C)1513 2021 y Fl(\()p Fk(C)1628 1985 y Fg(0)1651 2021 y Fk(;)17 b Fr(-)o Fl(\))31 b(:)g Fm(C)37 b(\000)-59 b(!)31 b Fo(Set)j Fr(b)s(e)h(a)f(represen)m(table)i(functor.)49 b(By)0 2137 y(the)40 b(de\034nition)e(of)g(the)i(limit)c(of)i Fm(F)49 b Fr(there)40 b(is)f(a)f(unique)i(morphism)d Fk(f)50 b Fl(:)39 b Fk(C)2860 2101 y Fg(0)2922 2137 y Fm(\000)-59 b(!)38 b Fl(lim)3078 2183 y Fm( )-42 b(\000)3214 2137 y Fl(\()p Fm(F)10 b Fl(\))38 b Fr(with)0 2253 y Fk(\031)t Fm(K)135 2268 y Fj(f)208 2253 y Fl(=)27 b Fk(')j Fr(for)f(eac)m(h)h(natural)e(transformation)f Fk(')h Fl(:)g Fm(K)1989 2268 y Fj(C)2044 2249 y Ff(0)2098 2253 y Fm(\000)-60 b(!)28 b(F)10 b Fr(.)41 b(This)30 b(de\034nes)h(an)f (isomorphism)0 2370 y Fl(Nat\()p Fm(K)274 2385 y Fj(C)329 2366 y Ff(0)355 2370 y Fk(;)17 b Fm(F)10 b Fl(\))546 2342 y Fm(\030)547 2374 y Fl(=)651 2370 y(Mor)827 2385 y Fg(C)872 2370 y Fl(\()p Fk(C)987 2333 y Fg(0)1011 2370 y Fk(;)17 b Fl(lim)1055 2415 y Fm( )-42 b(\000)1190 2370 y Fl(\()p Fm(F)10 b Fl(\)\))p Fr(.)42 b(Hence)34 b(w)m(e)g(ha)m(v)m(e) 264 2536 y Fl(lim)264 2582 y Fm( )-42 b(\000)399 2536 y Fl(\(Mor)613 2551 y Fg(C)658 2536 y Fl(\()p Fk(C)r(;)17 b Fm(F)10 b Fl(\)\))997 2509 y Fm(\030)997 2541 y Fl(=)264 2654 y Fm(f)p Fl(\()p Fk(')o Fl(\()p Fk(D)s Fl(\))28 b(:)f Fk(C)734 2618 y Fg(0)785 2654 y Fm(\000)-59 b(!)27 b(F)10 b Fl(\()p Fk(D)s Fl(\))p Fm(j)p Fk(D)29 b Fm(2)f(D)s Fl(\))p Fm(j8)p Fl(\()p Fk(f)38 b Fl(:)28 b Fk(D)i Fm(\000)-59 b(!)27 b Fk(D)2124 2618 y Fg(0)2147 2654 y Fl(\))h Fm(2)g(D)i Fl(:)e Fm(F)10 b Fl(\()p Fk(f)h Fl(\))p Fk(')o Fl(\()p Fk(D)s Fl(\))27 b(=)h Fk(')p Fl(\()p Fk(D)3226 2618 y Fg(0)3248 2654 y Fl(\))p Fm(g)264 2771 y Fl(=)f(Nat\()p Fm(K)641 2786 y Fj(C)696 2767 y Ff(0)723 2771 y Fk(;)17 b Fm(F)10 b Fl(\))913 2743 y Fm(\030)914 2775 y Fl(=)1019 2771 y(Mor)1195 2786 y Fg(C)1240 2771 y Fl(\()p Fk(C)1355 2734 y Fg(0)1378 2771 y Fk(;)17 b Fl(lim)1422 2816 y Fm( )-42 b(\000)1557 2771 y Fl(\()p Fm(F)10 b Fl(\)\))p Fk(:)p 1821 2783 34 63 v 149 2947 a Fo(Corollary)36 b(8.10.13.)49 b Fn(L)-5 b(et)45 b Fm(F)56 b Fl(:)45 b Fm(C)52 b(\000)-57 b(!)46 b(C)1775 2910 y Fg(0)1843 2947 y Fn(b)-5 b(e)44 b(left)h(adjoint)f(to)g Fm(G)52 b Fl(:)46 b Fm(C)2858 2910 y Fg(0)2927 2947 y Fm(\000)-57 b(!)46 b(C)6 b Fn(.)74 b(Then)44 b Fm(F)0 3063 y Fn(pr)-5 b(eserves)34 b(c)-5 b(olimits)34 b(and)g Fm(G)41 b Fn(pr)-5 b(eserves)34 b(limits.)149 3237 y Fa(Pr)m(oof.)42 b Fr(F)-8 b(or)31 b(a)i(diagram)d Fm(H)f Fl(:)f Fm(D)i(\000)-60 b(!)28 b(C)38 b Fr(w)m(e)c(ha)m(v)m(e)552 3387 y Fl(Mor)728 3402 y Fg(C)773 3387 y Fl(\()p Fr(-)p Fk(;)17 b Fl(lim)888 3433 y Fm( )-42 b(\000)1023 3387 y Fl(\()p Fm(G)6 b(H)q Fl(\)\))1315 3359 y Fm(\030)1316 3391 y Fl(=)1420 3387 y(lim)1420 3433 y Fm( )-42 b(\000)1572 3387 y Fl(Mor)1748 3402 y Fg(C)1793 3387 y Fl(\()p Fr(-)p Fk(;)17 b Fm(G)6 b(H)q Fl(\))2123 3359 y Fm(\030)2124 3391 y Fl(=)2228 3387 y(lim)2228 3433 y Fm( )-42 b(\000)2380 3387 y Fl(Mor)2557 3402 y Fg(C)2598 3383 y Ff(0)2624 3387 y Fl(\()p Fm(F)10 b Fr(-)o Fk(;)17 b Fm(H)q Fl(\))2970 3359 y Fm(\030)2971 3391 y Fl(=)928 3505 y(Mor)1104 3520 y Fg(C)1145 3501 y Ff(0)1171 3505 y Fl(\()p Fm(F)10 b Fr(-)o Fk(;)17 b Fl(lim)1367 3551 y Fm( )-42 b(\000)1503 3505 y Fl(\()p Fm(H)q Fl(\)\))1729 3477 y Fm(\030)1730 3509 y Fl(=)1834 3505 y(Mor)2010 3520 y Fg(C)2055 3505 y Fl(\()p Fr(-)p Fk(;)17 b Fm(G)6 b Fl(\(lim)2273 3551 y Fm( )-42 b(\000)2408 3505 y Fl(\()p Fm(H)q Fl(\)\)\))p Fk(;)0 3665 y Fr(hence)39 b Fl(lim)276 3711 y Fm( )-42 b(\000)412 3665 y Fl(\()p Fm(G)6 b(H)q Fl(\))675 3637 y Fm(\030)675 3669 y Fl(=)789 3665 y Fm(G)g Fl(\(lim)892 3711 y Fm( )-42 b(\000)1027 3665 y Fl(\()p Fm(H)q Fl(\)\))38 b Fr(as)g(represen)m(ting)h(ob)5 b(jects.)61 b(The)38 b(pro)s(of)f(for)h(the)g(left)f(adjoin)m(t)0 3781 y(functor)c(is)f(analogous.)p 923 3793 V eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF