reserve C for category,
  o1, o2, o3 for Object of C;

theorem Th14:
  for C1, C2 being non empty AltCatStr for F being Contravariant
FunctorStr over C1, C2 holds F is full iff for o1, o2 being Object of C1 holds
  Morph-Map(F,o2,o1) is onto
proof
  let C1, C2 be non empty AltCatStr, F be Contravariant FunctorStr over C1, C2;
  set I = [:the carrier of C1, the carrier of C1:];
  hereby
    assume
A1: F is full;
    let o1, o2 be Object of C1;
    thus Morph-Map(F,o2,o1) is onto
    proof
A2:   [o2,o1] in I by ZFMISC_1:87;
      then
A3:   [o2,o1] in dom(the ObjectMap of F) by FUNCT_2:def 1;
      consider f being ManySortedFunction of the Arrows of C1, (the Arrows of
      C2)*the ObjectMap of F such that
A4:   f = the MorphMap of F and
A5:   f is "onto" by A1;
      rng(f.[o2,o1]) = ((the Arrows of C2)*the ObjectMap of F).[o2,o1] by A5,A2
;
      hence
      rng(Morph-Map(F,o2,o1)) = (the Arrows of C2).((the ObjectMap of F).
      (o2,o1)) by A4,A3,FUNCT_1:13
        .= <^F.o1,F.o2^> by FUNCTOR0:23;
    end;
  end;
  assume
A6: for o1,o2 being Object of C1 holds Morph-Map(F,o2,o1) is onto;
  ex I29 being non empty set, B9 being ManySortedSet of I29, f9 being
  Function of I, I29 st the ObjectMap of F = f9 & the Arrows of C2 = B9 & the
MorphMap of F is ManySortedFunction of the Arrows of C1, B9*f9 by
FUNCTOR0:def 3;
  then reconsider
  f = the MorphMap of F as ManySortedFunction of the Arrows of C1,
  (the Arrows of C2)*the ObjectMap of F;
  take f;
  thus f = the MorphMap of F;
  let i be set;
  assume i in I;
  then consider o2, o1 being object such that
A7: o2 in the carrier of C1 & o1 in the carrier of C1 and
A8: i = [o2,o1] by ZFMISC_1:84;
  reconsider o1, o2 as Object of C1 by A7;
  [o2,o1] in I by ZFMISC_1:87;
  then
A9: [o2,o1] in dom(the ObjectMap of F) by FUNCT_2:def 1;
  Morph-Map(F,o2,o1) is onto by A6;
  then rng(Morph-Map(F,o2,o1)) = (the Arrows of C2).(F.o1,F.o2)
    .= (the Arrows of C2).((the ObjectMap of F).(o2,o1)) by FUNCTOR0:23
    .= ((the Arrows of C2)*the ObjectMap of F).[o2,o1] by A9,FUNCT_1:13;
  hence thesis by A8;
end;
