reserve C for CategoryStr;
reserve f,f1,f2,f3 for morphism of C;
reserve g1,g2 for morphism of C opp;
reserve C,D,E for with_identities CategoryStr;
reserve F for Functor of C,D;
reserve G for Functor of D,E;
reserve f for morphism of C;

theorem
  for C,D being non empty category, F being covariant Functor of C,D
  holds F is Functor of Alter(C), Alter(D)
  proof
    let C,D be non empty category;
    let F be covariant Functor of C,D;
    reconsider F1 = F as
    Function of the carrier' of Alter(C),the carrier' of Alter(D);
A1: F is identity-preserving & F is multiplicative by Def25;
A2: for a being Object of Alter(C) ex b being Object of Alter(D)
    st F1.(id a) = id b
    proof
      let a be Object of Alter(C);
      reconsider a1 = id a as morphism of C;
      a1 is identity by Th47;
      then consider b be Object of Alter(D) such that
A3:   F.a1 = id b by Th47,A1;
      take b;
      thus F1.(id a) = id b by A3,Def21;
    end;
A4: for f being Morphism of Alter(C) holds F1.(id dom f) = id dom (F1.f) &
    F1.(id cod f) = id cod (F1.f)
    proof
      let f be Morphism of Alter(C);
      reconsider o1 = dom f as Object of Alter(C);
      reconsider o2 = o1 as Object of C;
      reconsider f1=f as morphism of C;
A5:   F.f1 = F1.f by Def21;
A6:   F.o2 = F1.(dom f1) by Th45
      .= dom(F.f1) by Th32
      .= dom(F1.f) by Th45,A5;
      thus F1.(id dom f) = F1.(id- o2) by Th46
      .= id-(F.o2)
      .= id dom (F1.f) by A6,Th46;
      reconsider o3 = cod f as Object of Alter(C);
      reconsider o4 = o3 as Object of C;
      reconsider f1=f as morphism of C;
A7:   F.f1 = F1.f by Def21;
A8:   F.o3 = F1.(cod f1) by Th45
      .= cod(F.f1) by Th32
      .= cod(F1.f) by Th45,A7;
      thus F1.(id cod f) = F1.(id- o4) by Th46
      .= id-(F.o4)
      .= id cod (F1.f) by A8,Th46;
    end;
    for f,g being Morphism of Alter(C) st dom g = cod f
    holds F1.(g(*)f) = (F1.g)(*)(F1.f)
    proof
      let f,g be Morphism of Alter(C);
      assume
A9:   dom g = cod f;
      reconsider f1=f,g1=g as morphism of C;
A10:   [g1,f1] in dom the composition of C by A9,CAT_1:15;
A11:   g1 |> f1 by A9,CAT_1:15;
A12:   (the composition of C).(g1,f1) = g1(*)f1 by A10,Def2,Def3;
A13:   F.g1 = F1.g & F.f1 = F1.f by Def21;
A14:   [F1.g,F1.f] in dom the Comp of Alter D by A13,Def2,A11,A1;
      thus F1.(g(*)f) = F1.((the Comp of Alter C).(g,f)) by A9,CAT_1:16
      .= F.(g1(*)f1) by A12,Def21
      .= (F.g1)(*)(F.f1) by A11,A1
      .= (the Comp of Alter(D)).(F1.g,F1.f) by A13,Def3,A11,A1
      .= (F1.g)(*)(F1.f) by A14,CAT_1:def 1;
    end;
    hence thesis by A2,A4,CAT_1:61;
  end;
