reserve B,C,D,C9,D9 for Category;
reserve E for Subcategory of C;

theorem
  for c,c9 being Object of C, f being Morphism of c,c9, d,d9 being
  Object of D, g being Morphism of d,d9 st Hom(c,c9) <> {} & Hom(d,d9) <> {}
  holds [f,g] is Morphism of [c,d],[c9,d9]
proof
  let c,c9 be Object of C, f be Morphism of c,c9, d,d9 be Object of D, g be
  Morphism of d,d9;
  assume
A1: Hom(c,c9) <> {} & Hom(d,d9) <> {};
  then cod f = c9 & cod g = d9 by CAT_1:5;
  then
A2: cod [f,g] = [c9,d9] by Th22;
  dom f = c & dom g = d by A1,CAT_1:5;
  then dom [f,g] = [c,d] by Th22;
  hence thesis by A2,CAT_1:4;
end;
