reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;

theorem
  F is isomorphic implies (Obj F)" = Obj F"
proof
  assume
A1: F is isomorphic;
  then
A2: F is one-to-one;
A3: rng Obj F = the carrier of B by A1;
  then reconsider
  G = (Obj F)" as Function of the carrier of B, the carrier of A by A2,Th7,
FUNCT_2:25;
A4: Obj F is one-to-one by A2,Th7;
  now
    let b be Object of B;
    F.(id(G.b) qua Morphism of A) = id((Obj F).(G.b)) by CAT_1:68
      .= id b by A3,A4,FUNCT_1:35;
    then
    id(G.b) = (F qua Function of the carrier' of A, the carrier' of B)".id
    b by A2,FUNCT_2:26
      .= F".(id b qua Morphism of B) by A1,Def2;
    hence G.b = (Obj F").b by CAT_1:67;
  end;
  hence thesis by FUNCT_2:63;
end;
