reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;

theorem
  [:1Cat(o,m),A:] ~= A
proof
  take F = pr2(1Cat(o,m),A);
  set X = [:the carrier' of 1Cat(o,m), the carrier' of A:];
  now
    let x1,x2 be object;
    assume x1 in X;
    then consider x11,x12 being object such that
A1: x11 in the carrier' of 1Cat(o,m) and
A2: x12 in the carrier' of A and
A3: x1 = [x11,x12] by ZFMISC_1:def 2;
    assume x2 in X;
    then consider x21,x22 being object such that
A4: x21 in the carrier' of 1Cat(o,m) and
A5: x22 in the carrier' of A and
A6: x2 = [x21,x22] by ZFMISC_1:def 2;
    reconsider f11 = x11, f21 = x21 as Morphism of 1Cat(o,m) by A1,A4;
    assume
A7: F.x1 = F.x2;
    reconsider f12 = x12, f22 = x22 as Morphism of A by A2,A5;
A8: f11 = m by TARSKI:def 1
      .= f21 by TARSKI:def 1;
    f12 = F.(f11,f12) by FUNCT_3:def 5
      .= F.(f21,f22) by A3,A6,A7
      .= f22 by FUNCT_3:def 5;
    hence x1 = x2 by A3,A6,A8;
  end;
  hence F is one-to-one by FUNCT_2:19;
  thus thesis by FUNCT_3:46;
end;
