reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th2:
  the Comp of 1Cat(o,m) = {[[m,m],m]}
proof
  set A = 1Cat(o,m);
  reconsider f = m as Morphism of A by TARSKI:def 1;
  set a = the Object of A;
  thus the Comp of A c= {[[m,m],m]}
  proof
    set o9 = the Object of A;
    let x be object;
A1: dom id o9 = o9;
A2: cod id o9 = o9;
    assume
A3: x in the Comp of A;
    then consider x1,x2 being object such that
A4: x = [x1,x2] by RELAT_1:def 1;
A5: x1 in dom the Comp of A by A3,A4,XTUPLE_0:def 12;
    dom the Comp of A c= [:the carrier' of A, the carrier' of A:] by
RELAT_1:def 18;
    then consider x11,x12 being object such that
A6: x11 in the carrier' of A and
A7: x12 in the carrier' of A and
A8: x1 = [x11,x12] by A5,ZFMISC_1:def 2;
A9: x12 = id o9 by A7,ZFMISC_1:def 10;
A10: x2 is set by TARSKI:1;
    x11 = id o9 by A6,ZFMISC_1:def 10;
    then x2 = (the Comp of A).(id o9,id o9) by A3,A4,A5,A8,A9,FUNCT_1:def 2,A10
;
    then x2 = id o9(*)(id o9 qua Morphism of A) by A1,A2,CAT_1:16;
    then
A11: x2 = m by TARSKI:def 1;
A12: x12 = m by A7,TARSKI:def 1;
    x11 = m by A6,TARSKI:def 1;
    hence thesis by A4,A8,A12,A11,TARSKI:def 1;
  end;
  let x be object;
  assume x in {[[m,m],m]};
  then
A13: x = [[m,m],m] by TARSKI:def 1;
  f = id a by TARSKI:def 1;
  hence thesis by A13,Th1;
end;
