reserve x,y for set;

theorem Th30:
  for A,B being category, C being non empty subcategory of A for F
being contravariant Functor of A,B for a,b being Object of A, c,d being Object
  of C st c = a & d = b & <^c,d^> <> {} for f being Morphism of a,b for g being
  Morphism of c,d st g = f holds (F|C).g = F.f
proof
  let A,B be category, C be non empty subcategory of A;
  let F be contravariant Functor of A,B;
  let a,b be Object of A;
  let c,d be Object of C;
  assume that
A1: c = a & d = b and
A2: <^c,d^> <> {};
  let f be Morphism of a,b;
  let g be Morphism of c,d;
  assume g = f;
  then
A3: (incl C).g = f by A2,Th27;
  (incl C).c = a & (incl C).d = b by A1,FUNCTOR0:27;
  hence thesis by A2,A3,FUNCTOR3:8;
end;
