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;
reserve t for natural_transformation of F,F1,
  t1 for natural_transformation of F1,F2;
reserve a,b for Element of C;

theorem Th34:
  for A being discrete Category, B being Category, F1,F2 being
  Functor of B,A st F1 is_transformable_to F2 holds F1 = F2
proof
  let A be discrete Category, B be Category, F1,F2 be Functor of B,A;
  assume
A1: F1 is_transformable_to F2;
  now
    let m be Morphism of B;
    Hom(F1.dom m,F2.dom m) <> {} by A1;
    then
A2: F1.dom m = F2.dom m by Def17;
A3: m in Hom(dom m,cod m);
    then Hom(F1.dom m, F1.cod m) <> {} by CAT_1:81;
    then F1.dom m = F1.cod m by Def17;
    then
A4: Hom(F1.dom m, F1.cod m) = { id(F1.dom m) } by Th33;
    Hom(F2.dom m, F2.cod m) <> {} by A3,CAT_1:81;
    then F2.dom m = F2.cod m by Def17;
    then
A5: Hom(F2.dom m, F2.cod m) = { id(F2.dom m) } by Th33;
    F1.m in Hom(F1.dom m, F1.cod m) by A3,CAT_1:81;
    then
A6: F1.m = id(F1.dom m) by A4,TARSKI:def 1;
    F2.m in Hom(F2.dom m,F2.cod m) by A3,CAT_1:81;
    hence F1.m = F2.m by A2,A6,A5,TARSKI:def 1;
  end;
  hence thesis;
end;
