reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem Th41:
  for F being Functor of A,B, G being Functor of C,D for I,J being
  Functor of B,C st I ~= J holds G*I ~= G*J & I*F ~= J*F
proof
  let F be Functor of A,B, G be Functor of C,D;
  let I,J be Functor of B,C;
  assume
A1: I is_naturally_transformable_to J;
  given t being natural_transformation of I,J such that
A2: t is invertible;
  thus G*I ~= G*J
  proof
    thus G*I is_naturally_transformable_to G*J by A1,Th20;
    take G*t;
    let b be Object of B;
A3: t.b is invertible by A2;
A4:  G.(I.b) = (G*I).b & G.(J.b) = (G*J).b by CAT_1:76;
  (G*t).b = G/.(t.b) by A1,Th21;
    hence (G*t).b is invertible by A3,Th1,A4;
  end;
  thus I*F is_naturally_transformable_to J*F by A1,Th20;
  take t*F;
  let a be Object of A;
A5:  I.(F.a) = (I*F).a & J.(F.a) = (J*F).a by CAT_1:76;
 (t*F).a = t.(F.a) by A1,Th22;
  hence (t*F).a is invertible by A2,A5;
end;
