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Fm(I)8 b Fs(\))22 b Fn(\012)h Fs(\()p Fm(A)f Fn(\012)h Fm(B)5 b Fs(\))1568 b Fm(I)30 b Fn(\012)23 b Fs(\()p Fm(I)30 b Fn(\012)23 b Fs(\()p Fm(A)f Fn(\012)g Fm(B)5 b Fs(\)\))p 1044 3341 1512 4 v 2473 3339 a Fl(-)1777 3320 y Fk(\013)984 3160 y(\032)p Fi(\012)p Fj(\(1)p Fi(\012)p Fj(1\))732 3271 y Fl(\021)815 3215 y(\021)898 3160 y(\021)982 3105 y(\021)1030 3072 y(\021)-83 b(3)2485 3163 y Fj(1)p Fi(\012)p Fk(\025)2487 3072 y Fl(Q)2570 3128 y(Q)2653 3183 y(Q)2736 3238 y(Q)2785 3271 y(Q)g(s)0 3519 y Fs(all)21 b(sub)s(diagrams)h(comm)m(ute)g(except) j(for)d(the)h(righ)m(t)g(hand)g(trap)s(ezoid.)39 b(Since)23 b(all)e(morphisms)h(are)0 3635 y(isomorphisms)32 b(the)h(righ)m(t)g (hand)h(trap)s(ezoid)f(m)m(ust)g(comm)m(ute)g(also.)45 b(Hence)35 b(the)f(\014rst)g(diagram)0 3751 y(of)e(the)h(Lemma)e(comm)m (utes.)149 3868 y(In)i(a)g(similar)c(w)m(a)m(y)34 b(one)f(sho)m(ws)h (the)f(comm)m(utativit)m(y)e(of)h(the)h(second)g(diagram.)149 3984 y(F)-8 b(urthermore)32 b(the)h(follo)m(wing)d(diagram)h(comm)m (utes)784 4195 y Fm(I)f Fn(\012)23 b Fs(\()p Fm(I)30 b Fn(\012)22 b Fm(I)8 b Fs(\))308 b(\()p Fm(I)30 b Fn(\012)23 b Fm(I)8 b Fs(\))22 b Fn(\012)g Fm(I)p 1285 4172 250 4 v 1285 4170 a Fl(\033)1387 4152 y Fk(\013)2344 4195 y Fm(I)30 b Fn(\012)23 b Fs(\()p Fm(I)30 b Fn(\012)22 b Fm(I)8 b Fs(\))p 2065 4172 V 2232 4170 a Fl(-)2167 4152 y Fk(\013)1298 4590 y Fm(I)30 b Fn(\012)23 b Fm(I)1028 4380 y Fj(1)p Fi(\012)p Fk(\032)1088 4322 y Fl(@)1171 4405 y(@)1254 4488 y(@)1259 4492 y(@)-83 b(R)1665 4373 y Fk(\032)1649 4322 y Fl(\000)1566 4405 y(\000)1483 4488 y(\000)1478 4492 y(\000)g(\011)2078 4590 y Fm(I)30 b Fn(\012)23 b Fm(I)1808 4380 y Fk(\032)p Fi(\012)p Fj(1)1868 4322 y Fl(@)1951 4405 y(@)2034 4488 y(@)2039 4492 y(@)-83 b(R)2445 4385 y Fj(1)p Fi(\012)p Fk(\025)2429 4322 y Fl(\000)2346 4405 y(\000)2263 4488 y(\000)2258 4492 y(\000)g(\011)1775 4984 y Fm(I)1508 4763 y Fk(\032)1478 4712 y Fl(@)1561 4795 y(@)1644 4878 y(@)1649 4882 y(@)g(R)2055 4763 y Fk(\032)2039 4712 y Fl(\000)1956 4795 y(\000)1873 4878 y(\000)1868 4882 y(\000)g(\011)149 5099 y Fs(Here)40 b(the)e(left)g(hand)h(triangle)d(comm)m(utes)j(b)m(y)g(the)g(previous)g (prop)s(ert)m(y)-8 b(.)61 b(The)40 b(comm)m(uta-)0 5216 y(tivit)m(y)34 b(of)f(the)i(righ)m(t)e(hand)i(diagram)d(is)i(giv)m(en)g (b)m(y)h(the)g(axiom.)47 b(The)35 b(lo)m(w)m(er)g(square)g(comm)m(utes) p eop %%Page: 80 6 80 5 bop 0 236 a Fq(80)333 b(3.)76 b(HOPF)33 b(ALGEBRAS,)g(ALGEBRAIC,)h (F)n(ORMAL,)e(AND)h(QUANTUM)f(GR)n(OUPS)0 425 y Fs(since)j Fm(\032)g Fs(is)f(a)g(natural)f(transformation.)47 b(In)35 b(particular)e Fm(\032)p Fs(\(1)23 b Fn(\012)h Fm(\032)p Fs(\))31 b(=)g Fm(\032)p Fs(\(1)24 b Fn(\012)g Fm(\025)p Fs(\).)49 b(Since)34 b Fm(\032)h Fs(is)f(an)0 541 y(isomorphism)c(and)j Fm(I)d Fn(\012)22 b Fs(-)993 514 y Fn(\030)993 546 y Fs(=)1098 541 y(Id)1187 556 y Fi(C)1265 541 y Fs(w)m(e)33 b(get)g Fm(\032)28 b Fs(=)f Fm(\025)p Fs(.)p 3529 541 4 66 v 3533 479 59 4 v 3533 541 V 3591 541 4 66 v 149 724 a Fp(Problem)36 b(3.2.2.)49 b Fs(F)-8 b(or)45 b(morphisms)f Fm(f)60 b Fs(:)50 b Fm(I)57 b Fn(\000)-60 b(!)50 b Fm(M)56 b Fs(and)45 b Fm(g)53 b Fs(:)c Fm(I)58 b Fn(\000)-60 b(!)49 b Fm(N)56 b Fs(in)45 b(a)g(monoidal)0 840 y(category)j(w)m(e)i (de\014ne)f(\()p Fm(f)44 b Fn(\012)33 b Fs(1)54 b(:)g Fm(N)65 b Fn(\000)-60 b(!)54 b Fm(M)44 b Fn(\012)33 b Fm(N)10 b Fs(\))54 b(:=)h(\()p Fm(f)43 b Fn(\012)33 b Fs(1)2477 855 y Fk(I)2517 840 y Fs(\))p Fm(\032)p Fs(\()p Fm(I)8 b Fs(\))2732 804 y Fi(\000)p Fj(1)2874 840 y Fs(and)49 b(\(1)32 b Fn(\012)h Fm(g)58 b Fs(:)c Fm(M)0 957 y Fn(\000)-60 b(!)28 b Fm(M)33 b Fn(\012)22 b Fm(N)10 b Fs(\))28 b(:=)g(\(1)22 b Fn(\012)h Fm(g)t Fs(\))p Fm(\025)p Fs(\()p Fm(I)8 b Fs(\))1138 920 y Fi(\000)p Fj(1)1231 957 y Fs(.)43 b(Sho)m(w)34 b(that)e(the)h(diagram)1418 1641 y Fm(N)394 b(M)33 b Fn(\012)22 b Fm(N)p 1535 1613 326 4 v 1778 1611 a Fl(-)1632 1676 y Fk(f)7 b Fi(\012)p Fj(1)1437 1157 y Fm(I)515 b(M)p 1517 1125 450 4 v 1884 1123 a Fl(-)1720 1091 y Fk(f)p 1461 1542 4 351 v 1462 1542 a Fl(?)1387 1375 y Fk(g)p 2045 1542 V 2047 1542 a Fl(?)2086 1382 y Fj(1)p Fi(\012)p Fk(g)0 1814 y Fs(comm)m(utes.)149 1996 y Fp(De\014nition)37 b(3.2.6.)49 b Fs(Let)33 b(\()p Fn(C)6 b Fm(;)17 b Fn(\012)p Fs(\))32 b(and)h(\()p Fn(D)s Fm(;)17 b Fn(\012)p Fs(\))32 b(b)s(e)h(monoidal)d(categories.)43 b(A)33 b(functor)1563 2168 y Fn(F)k Fs(:)28 b Fn(C)34 b(\000)-60 b(!)27 b(D)0 2340 y Fs(together)33 b(with)f(a)g(natural)g(transformation)944 2511 y Fm(\030)5 b Fs(\()p Fm(M)g(;)17 b(N)10 b Fs(\))28 b(:)f Fn(F)10 b Fs(\()p Fm(M)g Fs(\))23 b Fn(\012)f(F)10 b Fs(\()p Fm(N)g Fs(\))28 b Fn(\000)-60 b(!)28 b(F)10 b Fs(\()p Fm(M)32 b Fn(\012)23 b Fm(N)10 b Fs(\))0 2683 y(and)33 b(a)f(morphism)1457 2814 y Fm(\030)1500 2829 y Fj(0)1567 2814 y Fs(:)27 b Fm(I)1664 2829 y Fi(D)1753 2814 y Fn(\000)-60 b(!)28 b(F)10 b Fs(\()p Fm(I)2061 2829 y Fi(C)2105 2814 y Fs(\))0 2965 y(is)32 b(called)f Fo(we)-5 b(akly)35 b(monoidal)c Fs(if)g(the)i(follo)m(wing)d(diagrams)h (comm)m(ute)178 3151 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