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
  Functors(1Cat(o,m),A) ~= A
proof
  reconsider a = o as Object of 1Cat(o,m) by TARSKI:def 1;
  take F = a |-> A;
  now
    let x,y be object;
A1: the carrier' of Functors(1Cat(o,m),A) = NatTrans(1Cat(o,m),A) by
NATTRA_1:def 17;
    assume x in the carrier' of Functors(1Cat(o,m),A);
    then consider F1,F2 being Functor of 1Cat(o,m),A, t being
    natural_transformation of F1,F2 such that
A2: x = [[F1,F2],t] and
A3: F1 is_naturally_transformable_to F2 by A1,NATTRA_1:def 16;
    assume y in the carrier' of Functors(1Cat(o,m),A);
    then consider F19,F29 being Functor of 1Cat(o,m),A, t9 being
    natural_transformation of F19,F29 such that
A4: y = [[F19,F29],t9] and
A5: F19 is_naturally_transformable_to F29 by A1,NATTRA_1:def 16;
    assume F.x = F.y;
    then
A6: t.a = F.y by A2,A3,Def1
      .= t9.a by A4,A5,Def1;
    reconsider G1=F1, G19=F19, G2=F2, G29=F29 as Function of {m}, the carrier'
    of A;
    reconsider s=t,s9=t9 as Function of {a}, the carrier' of A;
A7: id a = m by TARSKI:def 1;
A8: F1 is_transformable_to F2 by A3;
    then
A9: Hom(F1.a,F2.a) <> {};
A10: F19 is_transformable_to F29 by A5;
    then
A11: Hom(F19.a,F29.a) <> {};
    then F1.a = F19.a by A6,A9,CAT_1:6;
    then G1.id a = id(F19.a) by CAT_1:71
      .= G19.id a by CAT_1:71;
    then
A12: F1 = F19 by A7,FUNCT_2:125;
    F2.a = F29.a by A6,A9,A11,CAT_1:6;
    then G2.id a = id(F29.a) by CAT_1:71
      .= G29.id a by CAT_1:71;
    then
A13: F2 = F29 by A7,FUNCT_2:125;
    s.a = t9.a by A8,A6,NATTRA_1:def 5
      .= s9.a by A10,NATTRA_1:def 5;
    hence x = y by A2,A4,A12,A13,FUNCT_2:125;
  end;
  hence F is one-to-one by FUNCT_2:19;
  thus rng F c= the carrier' of A;
  let x be object;
  assume x in the carrier' of A;
  then reconsider f = x as Morphism of A;
  reconsider F1 = {m} --> id dom f, F2 = {m} --> id cod f as Function of the
  carrier' of 1Cat(o,m),the carrier' of A;
A14: now
    let g be Morphism of 1Cat(o,m);
    thus F1.(id dom g) = id dom f by FUNCOP_1:7
      .= id dom id dom f
      .= id dom (F1.g) by FUNCOP_1:7;
    thus F1.(id cod g) = id dom f by FUNCOP_1:7
      .= id cod id dom f
      .= id cod (F1.g) by FUNCOP_1:7;
  end;
A15: now
    let h,g be Morphism of 1Cat(o,m) such that
    dom g = cod h;
A16: Hom(dom f,dom f) <> {};
    thus F1.(g(*)h) = id dom f by FUNCOP_1:7
      .= (id dom f)*(id dom f)
      .= (id dom f)(*)(id dom f qua Morphism of A)by A16,CAT_1:def 13
      .= (id dom f)(*)(F1.h)by FUNCOP_1:7
      .= (F1.g)(*)(F1.h)by FUNCOP_1:7;
  end;
A17: now
    let h,g be Morphism of 1Cat(o,m) such that
    dom g = cod h;
A18: Hom(cod f,cod f) <> {};
    thus F2.(g(*)h) = id cod f by FUNCOP_1:7
      .= (id cod f)*(id cod f)
      .= (id cod f)(*)(id cod f qua Morphism of A)by A18,CAT_1:def 13
      .= (id cod f)(*)(F2.h)by FUNCOP_1:7
      .= (F2.g)(*)(F2.h)by FUNCOP_1:7;
  end;
A19: now
    let g be Morphism of 1Cat(o,m);
    thus F2.(id dom g) = id cod f by FUNCOP_1:7
      .= id dom id cod f
      .= id dom (F2.g) by FUNCOP_1:7;
    thus F2.(id cod g) = id cod f by FUNCOP_1:7
      .= id cod id cod f
      .= id cod (F2.g) by FUNCOP_1:7;
  end;
  ( for c being Object of 1Cat(o,m) ex d being Object of A st F1.(id c) =
id d)& for c being Object of 1Cat(o,m) ex d being Object of A st F2.(id c) = id
  d by FUNCOP_1:7;
  then reconsider F1, F2 as Functor of 1Cat(o,m),A by A14,A15,A19,A17,CAT_1:61;
  reconsider t = {a} --> f as Function of the carrier of 1Cat(o,m), the
  carrier' of A;
A20: for b being Object of 1Cat(o,m) holds F1.b = dom f & F2.b = cod f
  proof
    let b be Object of 1Cat(o,m);
    F2.(id b qua Morphism of 1Cat(o,m)) = id cod f by FUNCOP_1:7;
    then
A21: id(F2.b) = id cod f by CAT_1:71;
    F1.(id b qua Morphism of 1Cat(o,m)) = id dom f by FUNCOP_1:7;
    then id(F1.b) = id dom f by CAT_1:71;
    hence thesis by A21,CAT_1:59;
  end;
A22: now
    let b be Object of 1Cat(o,m);
A23: F2.b = cod f by A20;
    t.b = f & F1.b = dom f by A20,FUNCOP_1:7;
    then t.b in Hom(F1.b,F2.b) by A23;
    hence t.b is Morphism of F1.b,F2.b by CAT_1:def 5;
  end;
A24: now
    let b be Object of 1Cat(o,m);
    F1.b = dom f & F2.b = cod f by A20;
    hence Hom(F1.b,F2.b) <> {} by CAT_1:2;
  end;
  then
A25: F1 is_transformable_to F2;
  then reconsider t as transformation of F1,F2 by A22,NATTRA_1:def 3;
A26: for b being Object of 1Cat(o,m) holds t.b = f
  proof
    let b be Object of 1Cat(o,m);
    thus f = ({a} --> f).b by FUNCOP_1:7
      .= t.b by A25,NATTRA_1:def 5;
  end;
A27: now
    let b1,b2 be Object of 1Cat(o,m) such that
A28: Hom(b1,b2) <> {};
A29: Hom(F2.b1,F2.b2) <> {} by A28,CAT_1:82;
    let g be Morphism of b1,b2;
A30: t.b1 = f & Hom(F1.b1,F2.b1) <> {} by A24,A26;
A31: Hom(F1.b1,F1.b2) <> {} by A28,CAT_1:82;
A32: m in {m} by TARSKI:def 1;
A33: g = m by TARSKI:def 1;
    then
A34: F2/.g = F2.m by A28,CAT_3:def 10
      .= id cod f by A32,FUNCOP_1:7;
A35: F1/.g = F1.m by A28,A33,CAT_3:def 10
      .= id dom f by A32,FUNCOP_1:7;
    t.b2 = f & Hom(F1.b2,F2.b2) <> {} by A24,A26;
    hence t.b2*F1/.g = f(*)(F1/.g) by A31,CAT_1:def 13
      .= f by A35,CAT_1:22
      .= (F2/.g)(*)f by A34,CAT_1:21
      .= F2/.g*t.b1 by A30,A29,CAT_1:def 13;
  end;
  F1 is_transformable_to F2 by A24;
  then
A36: F1 is_naturally_transformable_to F2 by A27;
  then reconsider t as natural_transformation of F1,F2 by A27,NATTRA_1:def 8;
  [[F1,F2],t] in NatTrans(1Cat(o,m),A) by A36,NATTRA_1:def 16;
  then
A37: [[F1,F2],t] in the carrier' of Functors(1Cat(o,m),A) by NATTRA_1:def 17;
  F.[[F1,F2],t] = t.a by A36,Def1
    .= f by A26;
  hence thesis by A37,FUNCT_2:4;
end;
