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 F"" = F
proof
  reconsider f = F as Function of the carrier' of A, the carrier' of B;
  reconsider g = F" as Function of the carrier' of B, the carrier' of A;
  assume
A1: F is isomorphic;
  then
A2: F is one-to-one;
  thus F"" = g" by A1,Def2,Th8
    .= f"" by A1,Def2
    .= F by A2,FUNCT_1:43;
end;
