reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th18:
  for X being categorial non empty set, Y being non empty set
  for C1, C2 being strict Categorial Category st the carrier of C1 = X &
  (for A,B being Element of X, F being Functor of A,B holds
  [[A,B],F] is Morphism of C1 iff F in Y) & the carrier of C2 = X &
  (for A,B being Element of X, F being Functor of A,B holds
  [[A,B],F] is Morphism of C2 iff F in Y) holds C1 = C2
proof
  let X be categorial non empty set, Y be non empty set;
  let C1, C2 be strict Categorial Category such that
A1: the carrier of C1 = X and
A2: for A,B being Element of X, F being Functor of A,B holds
  [[A,B],F] is Morphism of C1 iff F in Y and
A3: the carrier of C2 = X and
A4: for A,B being Element of X, F being Functor of A,B holds
  [[A,B],F] is Morphism of C2 iff F in Y;
  the carrier' of C1 = the carrier' of C2
  proof
    hereby
      let x be object;
      assume x in the carrier' of C1;
      then reconsider m = x as Morphism of C1;
      reconsider a = dom m, b = cod m as Category by Th12;
      consider f being Functor of a,b such that
A5:   m = [[a,b],f] by Def6;
      f in Y by A1,A2,A5;
      then x is Morphism of C2 by A1,A4,A5;
      hence x in the carrier' of C2;
    end;
    let x be object;
    assume x in the carrier' of C2;
    then reconsider m = x as Morphism of C2;
    reconsider a = dom m, b = cod m as Category by Th12;
    consider f being Functor of a,b such that
A6: m = [[a,b],f] by Def6;
    f in Y by A3,A4,A6;
    then x is Morphism of C1 by A2,A3,A6;
    hence thesis;
  end;
  hence thesis by A1,A3,Th14;
end;
