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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 theSierpinski gasket, and their final co-algebras. The functors are defined on thecategories 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 therespective category of pointed metric spaces with the choice of continuous mapsis the final co-algebra of that with short maps and after forgetting the metricstructure is of that in the set setting. We use the fact that the finalco-algebra can be obtained by a Cauchy completion process, to construct themediating morphism from a co-algebra by means of the limit of a sequenceobtained by iterating the co-algebra. We also show that the Sierpinski gasket$~(\mathbb{S},\sigma)~$ is not the final co-algebra for these endofunctors whenthe morphism are restricted to being Lipschitz maps.

On the Initial Algebra and Final Co-algebra of some Endofunctors on Categories of Pointed Metric Spaces