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 A,B being Category, F being Functor of A,B
  holds F is covariant Functor of alter(A), alter(B)
  proof
    let A,B be Category;
    let F be Functor of A,B;
    reconsider F1 = F as Function of alter(A), alter(B);
    for f1,f2 being morphism of alter(A) st f1 |> f2
    holds F1.f1 |> F1.f2 & F1.(f1(*)f2) = (F1.f1)(*)(F1.f2)
    proof
      let f1,f2 be morphism of alter(A);
      assume
A1:  f1 |> f2;
      reconsider a1 = f1, a2 = f2 as Morphism of A;
A2:   dom a1 = cod a2 by A1,CAT_1:15;
      dom(F.a1) = F.(dom a1) & cod(F.a2) = F.(cod a2) by CAT_1:72;
      then dom(F.a1) = cod(F.a2) by A1,CAT_1:15;
      then
A3:   [F.a1,F.a2] in dom the Comp of B by CAT_1:15;
A4:   F1.f1 = F.a1 by Def21;
A5:   F1.f2 = F.a2 by Def21;
      thus F1.f1,F1.f2 are_composable by A3,A4,Def21;
      f1(*)f2 = a1(*)a2 by A1,Th41;
      hence F1.(f1(*)f2) = F.(a1(*)a2) by Def21
      .= (F.a1)(*)(F.a2) by A2,CAT_1:64
      .= (F1.f1)(*)(F1.f2) by A3,A4,A5,Th41;
    end;
    then
A6: F1 is multiplicative;
    for f being morphism of alter(A) st f is identity holds F1.f is identity
    proof
      let f be morphism of alter(A);
      assume f is identity;
      then consider o be Object of A such that
A7:  f = id o by Th42;
      consider o1 be Object of B such that
A8:  F.(id o) = id o1 by CAT_1:62;
      thus F1.f is identity by Def21,A7,A8,Th42;
    end;
    hence thesis by A6,Def25,Def22;
  end;
