
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 isomorphism
  holds P2 is isomorphism
  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 isomorphism;
    consider G1 be Functor of C,C1 such that
A5: G1 is covariant & G1 (*) F1 = id C1 & F1 (*) G1 = id C by A4;
    set G11 = G1 (*) F2;
    set G22 = id C2;
A6: G11 is covariant by A5,A1,CAT_6:35;
A7: F1 (*) G11 = (F1 (*) G1) (*) F2 by A5,A1,Th10
    .= F2 by A5,A1,Th11
    .= F2 (*) G22 by A1,Th11;
    consider Q2 be Functor of C2,D such that
A8: Q2 is covariant & P1 (*) Q2 = G11 & P2 (*) Q2 = G22
    & for H1 being Functor of C2,D st H1 is covariant & P1 (*) H1 = G11 &
    P2 (*) H1 = G22
    holds Q2 = H1 by A6,A2,A1,Def20,A7;
    set G33 = P1 (*) Q2 (*) P2;
A9: F2 (*) P2 is covariant by A1,CAT_6:35;
A10: G33 is covariant by A8,A6,A1,CAT_6:35;
    F1 (*) G33 = F1 (*) (G1 (*) (F2 (*) P2)) by A5,A8,A1,Th10
    .= (F1 (*) G1) (*) (F2 (*) P2) by A9,A5,A4,Th10
    .= F2 (*) P2 by A9,Th11,A5;
    then consider H be Functor of D,D such that
    H is covariant & P1 (*) H = G33 & P2 (*) H = P2 and
A11: for H1 being Functor of D,D st H1 is covariant & P1 (*) H1 = G33 &
     P2 (*) H1 = P2 holds H = H1
     by A1,A2,A10,Def20;
A12: P1 (*) id D = P1 by A1,Th11
    .= G1 (*) F1 (*) P1 by A1,A5,Th11
    .= G1 (*) (F1 (*) P1) by A1,A5,Th10
    .= G33 by A8,A3,A1,A5,Th10;
A13: P2 (*) id D = P2 by A1,Th11;
A14: P1 (*) (Q2 (*) P2) = G33 by A1,A8,Th10;
    P2 (*) (Q2 (*) P2) = P2 (*) Q2 (*) P2 by A1,A8,Th10
    .= P2 by A1,A8,Th11;
    then H = Q2 (*) P2 by A11,A14,A1,A8,CAT_6:35;
    hence P2 is isomorphism by A8,A13,A11,A12,A1;
  end;
