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 F1,F2 being Functor of A,B
   st for a,b being Object of A st Hom(a,b) <> {}
       for f being Morphism of a,b holds F1.f = F2.f
  holds F1 = F2
proof
  let F1,F2 be Functor of A,B such that
A1: for a,b being Object of A st Hom(a,b) <> {} for f being Morphism of
  a,b holds F1.f = F2.f;
  now
    let f be Morphism of A;
    reconsider f9 = f as Morphism of dom f, cod f by CAT_1:4;
    set a = dom f, b = cod f;
 Hom(dom f, cod f) <> {} by CAT_1:2;
    hence F1.f = F2.f9 by A1
      .= F2.f;
  end;
  hence thesis;
end;
