reserve y for set;
reserve C,D,E for Category,
  c,c1,c2 for Object of C,
  d,d1 for Object of D,
  x for set,
  f,f1 for (Morphism of E),
  g,g1 for (Morphism of C),
  h,h1 for (Morphism of D) ,
  F for Functor of C,E,
  G for Functor of D,E;
reserve o,o1,o2 for Element of commaObjs(F,G);

theorem Th1:
  (ex c,d,f st f in Hom(F.c,G.d)) implies o = [[o`11,o`12],o`2] & o
  `2 in Hom(F.o`11,G.o`12) & dom o`2 = F.o`11 & cod o`2 = G.o`12
proof
  assume ex c,d,f st f in Hom(F.c,G.d);
  then
A1: commaObjs(F,G) = {[[c,d],f]: f in Hom(F.c,G.d)} by Def1;
  o in commaObjs(F,G);
  then consider c,d,f such that
A2: o = [[c,d],f] and
A3: f in Hom(F.c,G.d) by A1;
  o`11 = c & o`12 = d by A2,MCART_1:85;
  hence thesis by A2,A3,CAT_1:1;
end;
