reserve A,B,C for Category,
  F,F1 for Functor of A,B;
reserve o,m for set;
reserve t for natural_transformation of F,F1;

theorem Th9:
  for a1,a2 being Object of A, b1,b2 being Object of B holds Hom(
  a1,a2) <> {} & Hom(b1,b2) <> {} iff Hom([a1,b1],[a2,b2]) <> {}
proof
  let a1,a2 be Object of A, b1,b2 be Object of B;
  Hom([a1,b1],[a2,b2]) = [:Hom(a1,a2),Hom(b1,b2):] by CAT_2:32;
  hence thesis by ZFMISC_1:90;
end;
