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

theorem Th37:
  for F1,G1 being Functor of A,B, F2,G2 being Functor of A,C st F1
  is_naturally_transformable_to G1 & F2 is_naturally_transformable_to G2 for t1
being natural_transformation of F1,G1, t2 being natural_transformation of F2,G2
  holds Pr1<:t1,t2:> = t1 & Pr2<:t1,t2:> = t2
proof
  let F1,G1 be Functor of A,B, F2,G2 be Functor of A,C such that
A1: F1 is_naturally_transformable_to G1 & F2 is_naturally_transformable_to G2;
A2: F1 is_transformable_to G1 & F2 is_transformable_to G2 by A1;
  let t1 be natural_transformation of F1,G1, t2 be natural_transformation of
  F2,G2;
  reconsider s = t1 as Function of the carrier of A, the carrier' of B;
  <:F1,F2:> is_naturally_transformable_to <:G1,G2:> by A1,Th36;
  then
A3: <:F1,F2:> is_transformable_to <:G1,G2:>;
  thus Pr1<:t1,t2:> = pr1(B,C) *(<:t1,t2:> qua transformation of <:F1,F2:>,<:
  G1,G2:>) by A1,Th36,ISOCAT_1:def 7
    .= pr1(B,C)* (<:t1,t2:> qua Function of the carrier of A, the carrier'
  of [:B,C:]) by A3,ISOCAT_1:def 5
    .= pr1(B,C)* (<:t1 qua transformation of F1,G1,t2:> qua Function of the
  carrier of A, the carrier' of [:B,C:]) by A1,Def12
    .= pr1(B,C)*<:s,t2:> by A2,Def11
    .= pr1(the carrier' of B, the carrier' of C)*<:s,t2:>
    .= t1 by FUNCT_3:62;
  thus Pr2<:t1,t2:> = pr2(B,C) *(<:t1,t2:> qua transformation of <:F1,F2:>,<:
  G1,G2:>) by A1,Th36,ISOCAT_1:def 7
    .= pr2(B,C)* (<:t1,t2:> qua Function of the carrier of A, the carrier'
  of [:B,C:]) by A3,ISOCAT_1:def 5
    .= pr2(B,C)* (<:t1 qua transformation of F1,G1,t2:> qua Function of the
  carrier of A, the carrier' of [:B,C:]) by A1,Def12
    .= pr2(B,C)*<:s,t2:> by A2,Def11
    .= pr2(the carrier' of B, the carrier' of C)*<:s,t2:>
    .= t2 by FUNCT_3:62;
end;
