
theorem
  ex C,C1,C2 being Category,
     F1 being Functor of C1,C, F2 being Functor of C2,C st
  not ex D being Category, P1 being Functor of D,C1,
         P2 being Functor of D,C2 st F1 * P1 = F2 * P2 &
  for D1 being Category, G1 being Functor of D1,C1,
   G2 being Functor of D1,C2 st F1 * G1 = F2 * G2 holds
  ex H being Functor of D1,D st P1 * H = G1 & P2 * H = G2
  & for H1 being Functor of D1,D st P1 * H1 = G1 & P2 * H1 = G2
    holds H = H1
  proof
    set C = Alter OrdC 2;
    set C1 = Alter OrdC 2;
    set C2 = Alter OrdC 2;
    consider f be morphism of OrdC 2 such that
    f is not identity and
    Ob OrdC 2 = {dom f, cod f} and
A1: Mor OrdC 2 = {dom f, cod f, f} and
A2: dom f, cod f, f are_mutually_distinct by Th39;
    reconsider g1 = dom f,g2 = cod f as morphism of OrdC 2
    by A1,ENUMSET1:def 1;
    reconsider F1 = MORPHISM(g1) as Functor of C1,C by CAT_6:47;
    reconsider F2 = MORPHISM(g2) as Functor of C2,C by CAT_6:47;
    take C,C1,C2,F1,F2;
    assume ex D being Category, P1 being Functor of D,C1,
    P2 being Functor of D,C2 st F1 * P1 = F2 * P2 & for D1 being Category,
    G1 being Functor of D1,C1, G2 being Functor of D1,C2 st F1 * G1 = F2 * G2
    holds ex H being Functor of D1,D st P1 * H = G1 & P2 * H = G2
    & for H1 being Functor of D1,D st P1 * H1 = G1 & P2 * H1 = G2 holds H = H1;
    then consider D be Category, P1 be Functor of D,C1,
    P2 be Functor of D,C2 such that
A3: F1 * P1 = F2 * P2 & for D1 being Category,
    G1 being Functor of D1,C1, G2 being Functor of D1,C2 st F1 * G1 = F2 * G2
    holds ex H being Functor of D1,D st P1 * H = G1 & P2 * H = G2
    & for H1 being Functor of D1,D st P1 * H1 = G1 & P2 * H1 = G2 holds H = H1;
    set g = the Morphism of D;
A4: g in the carrier' of D;
    then
A5: g in dom P1 by FUNCT_2:def 1;
A6: g in dom P2 by A4,FUNCT_2:def 1;
A7: Alter OrdC 2 = CatStr(# Ob OrdC 2,Mor OrdC 2,
    SourceMap OrdC 2, TargetMap OrdC 2,
    CompMap OrdC 2 #) by CAT_6:def 33;
    reconsider f1 = P1.g as morphism of OrdC 2 by A7;
    reconsider f2 = P2.g as morphism of OrdC 2 by A7;
A8: (F1 * P1).g = F1.(P1.g) by A5,FUNCT_1:13
    .= (MORPHISM g1).f1 by CAT_6:def 21
    .= g1 by Th40,CAT_6:22;
    (F2 * P2).g = F2.(P2.g) by A6,FUNCT_1:13
    .= (MORPHISM g2).f2 by CAT_6:def 21
    .= g2 by Th40,CAT_6:22;
    hence contradiction by A8,A3,A2,ZFMISC_1:def 5;
  end;
