reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  for o,m being object holds the Comp of 1Cat(o,m) = {[[m,m],m]}
  proof
    let o,m be object;
    set A = 1Cat(o,m);
A1: A = CatStr(# {o},{m},(m :-> o),(m :-> o),((m,m) :-> m) #)
      by CAT_1:def 11;
    then 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;
A2:   dom id o9 = o9;
A3:   cod id o9 = o9;
      assume
A4:   x in the Comp of A;
      then consider x1,x2 be object such that
A5:   x = [x1,x2] by RELAT_1:def 1;
A6:   x1 in dom the Comp of A by A4,A5,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 be object such that
A7:   x11 in the carrier' of A and
A8:   x12 in the carrier' of A and
A9:   x1 = [x11,x12] by A6,ZFMISC_1:def 2;
A10:  x12 = id o9 by A8,ZFMISC_1:def 10;
A11:  x2 is set by TARSKI:1;
      x11 = id o9 by A7,ZFMISC_1:def 10;
      then x2 = (the Comp of A).(id o9,id o9)
        by A4,A5,A6,A9,A10,FUNCT_1:def 2,A11;
      then x2 = id o9(*)(id o9 qua Morphism of A) by A2,A3,CAT_1:16;
      then
A12:  x2 = m by A1,TARSKI:def 1;
A13:  x12 = m by A1,A8,TARSKI:def 1;
      x11 = m by A1,A7,TARSKI:def 1;
      hence thesis by A5,A9,A13,A12,TARSKI:def 1;
    end;
    let x be object;
    assume x in {[[m,m],m]};
    then
A14: x = [[m,m],m] by TARSKI:def 1;
    f = id a by A1,TARSKI:def 1;
    hence thesis by A14,Th56;
  end;
