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);
reserve k,k1,k2,k9 for Element of commaMorphs(F,G);

theorem Th2:
  (ex c,d,f st f in Hom(F.c,G.d)) implies k = [[k`11,k`12], [k`21,k
`22]] & dom k`21 = k`11`11 & cod k`21 = k`12`11 & dom k`22 = k`11`12 & cod k`22
  = k`12`12 & (k`12`2)(*)(F.k`21) = (G.k`22)(*)(k`11`2)
proof
  assume ex c,d,f st f in Hom(F.c,G.d);
  then
A1: commaMorphs(F,G) = {[[o1,o2], [g,h]] : dom g = o1`11 & cod g = o2`11 &
  dom h = o1`12 & cod h = o2`12 & (o2`2)(*)(F.g) = (G.h)(*)(o1`2)} by Def2;
  k in commaMorphs(F,G);
  then consider g,h,o1,o2 such that
A2: k = [[o1,o2], [g,h]] and
A3: dom g = o1`11 & cod g = o2`11 & dom h = o1`12 & cod h = o2`12 & (o2
  `2)(*)(F.g) = (G.h)(*)(o1`2) by A1;
A4: k`21 = g by A2,MCART_1:85;
  k`11 = o1 & k`12 = o2 by A2,MCART_1:85;
  hence thesis by A2,A3,A4,MCART_1:85;
end;
