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

theorem Th21:
  for C being Categorial Category, D being full Categorial Category st
  the carrier of C c= the carrier of D holds C is Subcategory of D
proof
  let C be Categorial Category;
  let D be full Categorial Category;
  assume
A1: the carrier of C c= the carrier of D;
  the carrier' of C c= the carrier' of D
  proof
    let x be object;
    assume x in the carrier' of C;
    then reconsider x as Morphism of C;
    reconsider y1 = dom x, y2 = cod x as Category by Th12;
    consider F being Functor of y1,y2 such that
A2: x = [[y1,y2], F] by Def6;
A3: y1 is Object of D by A1;
    y2 is Object of D by A1;
    then [[y1,y2], F] is Morphism of D by A3,Def8;
    hence thesis by A2;
  end;
  hence thesis by Th15;
end;
