reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;
reserve t for natural_transformation of F,F1,
  t1 for natural_transformation of F1,F2;

theorem Th23:
  for A,B,F1,F2,t1 st F1 is_naturally_transformable_to F2 & t1 is
  invertible for a being Object of A holds t1".a = (t1.a)"
proof
  let A,B,F1,F2,t1 such that
A1: F1 is_naturally_transformable_to F2 and
A2: t1 is invertible;
  let a be Object of A;
A3: F1 is_transformable_to F2 by A1;
  thus t1".a = (t1 qua transformation of F1,F2)".a by A1,A2,Def12
    .= ((t1 qua transformation of F1,F2).a)" by A2,A3,Def11;
end;
