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

theorem Th20:
  for A being categorial non empty set
  ex C being full Categorial strict Category st the carrier of C = A
proof
  let AA be categorial non empty set;
  set dFF = the set of all
Funct(a,b) where a is Element of AA, b is Element of AA;
  set a = the Element of AA,f = the Functor of a,a;
A1: f in Funct(a,a) by CAT_2:def 2;
  Funct(a,a) in dFF;
  then reconsider FF = union dFF as non empty set by A1,TARSKI:def 4;
A2: now
    let A,B,C be Element of AA;
    let F be Functor of A,B, G be Functor of B,C;
    assume that F in FF and G in FF;
A3: G*F in Funct(A,C) by CAT_2:def 2;
    Funct(A,C) in dFF;
    hence G*F in FF by A3,TARSKI:def 4;
  end;
  now
    let A be Element of AA;
A4: id A in Funct(A,A) by CAT_2:def 2;
    Funct(A,A) in dFF;
    hence id A in FF by A4,TARSKI:def 4;
  end;
  then consider C being strict Categorial Category such that
A5: the carrier of C = AA and
A6: for A,B being Element of AA, F being Functor of A,B holds [[A,B],F]
  is Morphism of C iff F in FF
  by A2,Th17;
  C is full
  proof
    let a,b be Category;
    assume that
A7: a is Object of C and
A8: b is Object of C;
    reconsider A = a, B = b as Element of AA by A5,A7,A8;
    let F be Functor of a, b;
A9: F in Funct(A,B) by CAT_2:def 2;
    Funct(A,B) in dFF;
    then F in FF by A9,TARSKI:def 4;
    then [[A,B], F] is Morphism of C by A6;
    hence thesis;
  end;
  hence thesis by A5;
end;
