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 Th12:
  F is isomorphic & G is isomorphic implies G*F is isomorphic
proof
  assume that
A1: F is one-to-one and
A2: rng F = the carrier' of B and
A3: G is one-to-one and
A4: rng G = the carrier' of C;
  thus G*F is one-to-one by A1,A3,FUNCT_1:24;
  dom G = the carrier' of B by FUNCT_2:def 1;
  hence thesis by A2,A4,RELAT_1:28;
end;
