reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem
  for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of a,
  b holds (G*F)/.f = G/.(F/.f)
proof
  let a,b be Object of A;
  assume
A1: Hom(a,b) <> {};
  then
A2: Hom(F.a,F.b) <> {} by CAT_1:84;
  let f be Morphism of a,b;
  thus (G*F)/.f = (G*F).(f qua Morphism of A) by A1,CAT_3:def 10
    .= G.(F.(f qua Morphism of A)) by FUNCT_2:15
    .= G.(F/.f qua Morphism of B) by A1,CAT_3:def 10
    .= G/.(F/.f) by A2,CAT_3:def 10;
end;
