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 Th11:
  F is isomorphic implies F*F" = id B & F"*F = id A
proof
  reconsider f = F as Function of the carrier' of A, the carrier' of B;
A1: dom f = the carrier' of A by FUNCT_2:def 1;
  assume
A2: F is isomorphic;
  then
A3: F is one-to-one;
A4: rng f = the carrier' of B by A2;
  thus F*F" = f*f" by A2,Def2
    .= id B by A3,A4,FUNCT_1:39;
  thus F"*F = f"*f by A2,Def2
    .= id A by A3,A1,FUNCT_1:39;
end;
