  We consider two endofunctors of the form $~F:X\longrightarrow M\otimes X~$,
where $~M~$ is a non degenerate module, related to the unit interval and the
Sierpinski gasket, and their final co-algebras. The functors are defined on the
categories of bi-pointed and tri-pointed metric spaces, with continuous maps,
short maps or Lipschitz maps as the choice of morphisms.
  First we demonstrate that the final co-algebra for these endofunctors on the
respective category of pointed metric spaces with the choice of continuous maps
is the final co-algebra of that with short maps and after forgetting the metric
structure is of that in the set setting. We use the fact that the final
co-algebra can be obtained by a Cauchy completion process, to construct the
mediating morphism from a co-algebra by means of the limit of a sequence
obtained by iterating the co-algebra. We also show that the Sierpinski gasket
$~(\mathbb{S},\sigma)~$ is not the final co-algebra for these endofunctors when
the morphism are restricted to being Lipschitz maps.
