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
  for a,b being Object of Functors(A,B), f being Morphism of a,b st Hom(
  a,b) <> {} ex F,F1,t st a = F & b = F1 & f = [[F,F1],t]
proof
  let a,b be Object of Functors(A,B), f be Morphism of a,b such that
A1: Hom(a,b) <> {};
  the carrier' of Functors(A,B) = NatTrans(A,B) by Def16;
  then consider
  F1,F2 being Functor of A,B, t being natural_transformation of F1,F2
  such that
A2: f = [[F1,F2],t] and
  F1 is_naturally_transformable_to F2 by Def14;
  take F1,F2,t;
  thus a = dom f by A1,CAT_1:5
    .= F1 by A2,Th29;
  thus b = cod f by A1,CAT_1:5
    .= F2 by A2,Th29;
  thus thesis by A2;
end;
