
theorem
  for C,C1,C2,D being category,
      F1 being Functor of C1,C, F2 being Functor of C2,C,
      P1 being Functor of D,C1, P2 being Functor of D,C2
  st F1 is covariant & F2 is covariant & P1 is covariant & P2 is covariant &
     D,P1,P2 is_pullback_of F1,F2 & F1 is monomorphism
  holds P2 is monomorphism
  proof
    let C,C1,C2,D be category;
    let F1 be Functor of C1,C;
    let F2 be Functor of C2,C;
    let P1 be Functor of D,C1;
    let P2 be Functor of D,C2;
    assume
A1: F1 is covariant & F2 is covariant & P1 is covariant & P2 is covariant;
    assume
A2: D,P1,P2 is_pullback_of F1,F2;
    then
A3: F1 (*) P1 = F2 (*) P2 & for D1 being category,
    G1 being Functor of D1,C1, G2 being Functor of D1,C2
    st G1 is covariant & G2 is covariant & F1 (*) G1 = F2 (*) G2
    holds ex H being Functor of D1,D st H is covariant &
    P1 (*) H = G1 & P2 (*) H = G2
    & for H1 being Functor of D1,D st H1 is covariant & P1 (*) H1 = G1 &
    P2 (*) H1 = G2 holds H = H1 by A1,Def20;
    assume
A4: F1 is monomorphism;
    for D1 being category
    for Q1,Q2 being Functor of D1,D st Q1 is covariant & Q2 is covariant &
    P2 (*) Q1 = P2 (*) Q2 holds Q1 = Q2
    proof
      let D1 be category;
      let Q1,Q2 be Functor of D1,D;
      assume
A5:   Q1 is covariant & Q2 is covariant;
      assume
A6:   P2 (*) Q1 = P2 (*) Q2;
      set P11 = P1 (*) Q1;
      set P12 = P2 (*) Q1;
A7:   P11 is covariant & P12 is covariant by A1,A5,CAT_6:35;
      F1 (*) P11 = (F1 (*) P1) (*) Q1 by A5,A1,Th10
      .= F2 (*) P12 by A5,A3,A1,Th10;
      then consider H be Functor of D1,D such that
A8:   H is covariant & P1 (*) H = P11 & P2 (*) H = P12
      & for H1 being Functor of D1,D st H1 is covariant &
      P1 (*) H1 = P11 & P2 (*) H1 = P12 holds H = H1 by A1,A2,Def20,A7;
A9:   Q1 = H by A5,A8;
A10:  P1 (*) Q2 is covariant by A5,A1,CAT_6:35;
      F1 (*) (P1 (*) Q2) = (F1 (*) P1) (*) Q2 by A5,A1,Th10
      .= F2 (*) (P2 (*) Q2) by A5,A3,A1,Th10
      .= (F2 (*) P2) (*) Q1 by A6,A5,A1,Th10
      .= F1 (*) P11 by A5,A3,A1,Th10;
      hence Q1 = Q2 by A5,A9,A8,A6,A7,A10,A4;
    end;
    hence thesis by A1;
  end;
