reserve C for Category,
  C1,C2 for Subcategory of C;

theorem Th11:
  for C being Category holds 1Cat(C, [[C,C], id C]) is Categorial
proof
  let A be Category;
  set F = [[A,A], id A];
  set C = 1Cat(A, F);
  thus for x be Object of C holds x is Category by TARSKI:def 1;
  hereby
    let a be Object of C, D be Category;
    assume a = D;
    then D = A by TARSKI:def 1;
    hence id a = [[D,D], id D] by TARSKI:def 1;
  end;
  hereby
    let m be Morphism of C;
    let C1,C2 be Category;
    assume that
A1: C1 = dom m and
A2: C2 = cod m;
A3: C1 = A by A1,TARSKI:def 1;
A4: C2 = A by A2,TARSKI:def 1;
    reconsider G = id A as Functor of C1,C2 by A2,A3,TARSKI:def 1;
    take G;
    thus m = [[C1,C2],G] by A3,A4,TARSKI:def 1;
  end;
  let m1,m2 be Morphism of C;
  let A1,B,C be Category, F1 be Functor of A1,B, F2 be Functor of B,C;
  assume that
A5: m1 = [[A1,B],F1] and
A6: m2 = [[B,C],F2];
A7: [[A1,B],F1] = F by A5,TARSKI:def 1;
A8: [[B,C],F2] = F by A6,TARSKI:def 1;
A9: [A1,B] = [A,A] by A7,XTUPLE_0:1;
A10: [A,A] = [B,C] by A8,XTUPLE_0:1;
A11: F1 = id A by A7,XTUPLE_0:1;
A12: F2 = id A by A8,XTUPLE_0:1;
A13: A1 = A by A9,XTUPLE_0:1;
A14: C = A by A10,XTUPLE_0:1;
  F2*F1 = id A by A11,A12,FUNCT_2:17;
  hence thesis by A13,A14,TARSKI:def 1;
end;
