theorem Th59:
  for T being Functor of C,D for f,g being Morphism of C st dom g
  = cod f holds dom(T.g) = cod(T.f) & T.(g(*)f) = (T.g)(*)(T.f)
proof
  let T be Functor of C,D;
  let f,g be Morphism of C;
  assume
A1: dom g = cod f;
  then
A2: (the Comp of C).(g,f) = g(*)f & [g,f] in dom(the Comp of C)
     by Def4,Th14;
 id dom (T.g) = T.(id cod f) by A1,Def19
    .= id cod (T.f) by Def19;
  hence dom (T.g) = cod (T.f) by Th54;
  thus thesis by A2,Def19;
end;
