
theorem
  for C being Category, I being coIndexing of C for c1,c2 being Object
  of C st Hom(c1,c2) is non empty for m being Morphism of c1,c2 holds I`2.m is
  Functor of I`1.c2, I`1.c1
proof
  let C be Category, I be coIndexing of C;
  let c1,c2 be Object of C such that
A1: Hom(c1,c2) is non empty;
  let m be Morphism of c1,c2;
  dom (I`1*(the Source of C)) = the carrier' of C by PARTFUN1:def 2;
  then
A2: dom (I`1*(the Target of C)) = the carrier' of C & (I`1*(the Source of C)
  ).m = I`1.((the Source of C).m) by FUNCT_1:12,PARTFUN1:def 2;
  dom m = c1 & cod m = c2 by A1,CAT_1:5;
  hence thesis by A2,FUNCT_1:12;
end;
