
theorem Th24:
  for C1,C2 being Category, F being Functor of C1,C2 holds Upsilon
  F, Psi F form_morphism_between CatSign the carrier of C1, CatSign the carrier
  of C2
proof
  let C1,C2 be Category, F be Functor of C1,C2;
  set f = Upsilon F, g = Psi F;
  set S1 = CatSign the carrier of C1, S2 = CatSign the carrier of C2;
  thus dom f = the carrier of S1 & dom g = the carrier' of S1 by FUNCT_2:def 1;
  thus rng f c= the carrier of S2 & rng g c= the carrier' of S2;
  now
    let o be OperSymbol of CatSign the carrier of C1;
    per cases by Th16;
    suppose
      o`1 = 1;
      then consider a being Object of C1 such that
A1:   o = idsym(a) by Th17;
      thus (f*the ResultSort of S1).o = f.the_result_sort_of o by FUNCT_2:15
        .= f.homsym(a,a) by A1,Def3
        .= homsym(F.a,F.a) by Th21
        .= the_result_sort_of idsym(F.a) by Def3
        .= (the ResultSort of S2).(g.idsym a) by Th22
        .= ((the ResultSort of S2)*g).o by A1,FUNCT_2:15;
    end;
    suppose
      o`1 = 2;
      then consider a,b,c being Object of C1 such that
A2:   o = compsym(a,b,c) by Th18;
      thus (f*the ResultSort of S1).o = f.the_result_sort_of o by FUNCT_2:15
        .= f.homsym(a,c) by A2,Def3
        .= homsym(F.a,F.c) by Th21
        .= the_result_sort_of compsym(F.a,F.b,F.c) by Def3
        .= (the ResultSort of S2).(g.compsym(a,b,c)) by Th23
        .= ((the ResultSort of S2)*g).o by A2,FUNCT_2:15;
    end;
  end;
  hence f*the ResultSort of S1 = (the ResultSort of S2)*g;
  let o be set, p be Function;
  assume o in the carrier' of S1;
  then reconsider o9 = o as OperSymbol of S1;
  assume
A3: p = (the Arity of S1).o;
  per cases by Th16;
  suppose
    o9`1 = 1;
    then consider a being Object of C1 such that
A4: o = idsym(a) by Th17;
A5: f*{} = {};
    p = {} by A3,A4,Def3;
    hence f*p = the_arity_of idsym(F.a) by A5,Def3
      .= (the Arity of S2).(g.o) by A4,Th22;
  end;
  suppose
    o9`1 = 2;
    then consider a,b,c being Object of C1 such that
A6: o = compsym(a,b,c) by Th18;
    dom f = the carrier of S1 & p = <*homsym(b,c),homsym(a,b)*> by A3,A6,Def3,
FUNCT_2:def 1;
    hence f*p = <*f.homsym(b,c),f.homsym(a,b)*> by FINSEQ_2:125
      .= <*homsym(F.b,F.c),f.homsym(a,b)*> by Th21
      .= <*homsym(F.b,F.c),homsym(F.a,F.b)*> by Th21
      .= the_arity_of compsym(F.a,F.b,F.c) by Def3
      .= (the Arity of S2).(g.o) by A6,Th23;
  end;
end;
